Section Review — A / B / C
Review overview and remaining corrections
A — retain · B — condense · C — remove. Grade A means the material is worth preserving for the study structure, not that its mathematics has been certified. Core claims containing errors retain both the original text and revision guidance. Empty study sections are not filled in for the author.
2026-07-23-folland-measurable-structures
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| The Problem of Choosing Measurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. Set Systems | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 Rings | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Sigma-Rings | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.4 Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.5 Relations Between the Set Systems | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.6 What Each Structure Is Designed to Handle | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Generated Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.1 The Smallest Sigma-Algebra Containing a Family | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 Monotonicity of Generation | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.3 Working with Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.4 Preimages of Generated Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. Borel Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 Borel Sets on a Topological Space | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Standard Generators of $\mathcal B(\mathbb R)$ | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Countable Borel Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.4 How Topology Enters Measure Theory | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. Measurable Spaces and Maps | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 Measurable Spaces | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 Measurable Maps | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Checking Measurability on Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 5.1 Why We Do Not Measure Every Subset | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Why Generated Sigma-Algebras Are Useful | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 From Topological Sets to Measurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Introduction / Roadmap | C | An empty generic roadmap heading that duplicates the existing section order. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
2026-07-24-folland-measures-and-set-convergence
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| From Measurable Sets to Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. Measures and Measure Spaces | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 Definition of a Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 Measure Spaces | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Probability Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.4 Basic Examples | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Consequences of Countable Additivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.1 Finite Additivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 Monotonicity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.3 The Subtraction Formula | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.4 Countable Subadditivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. Disjointification of General Set Sequences | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 Why Countable Additivity Is Stated for Disjoint Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Converting a General Union into a Disjoint Union | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Applications of Disjointification | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. Convergence of Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 Increasing Sequences of Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 Continuity from Below | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Decreasing Sequences of Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.4 Continuity from Above | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.5 Why Finiteness Is Required from Above | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. Classes of Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.1 Finite Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Sigma-Finite Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 Semifinite Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.4 Relations Between the Finiteness Conditions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6. Null Sets and Almost-Everywhere Statements | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.1 Null Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.2 Almost-Everywhere Properties | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.3 Why Null Sets Matter in Analysis | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7. Complete Measures and Completion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.1 Complete Measure Spaces | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.2 Constructing the Completion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.3 Extension of the Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.4 Why Completeness Matters | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 8.1 Countable Additivity and Disjoint Decomposition | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.2 Set Convergence as Continuity of Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.3 Why Measure-Zero Sets Cannot Be Ignored Completely | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Introduction / Roadmap | C | An empty generic roadmap heading that duplicates the existing section order. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
2026-07-25-folland-caratheodory-extension
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| The Measure Extension Problem | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| The Central Construction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. The Extension Problem | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 What Is Initially Given? | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 What Must Be Constructed? | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Why Direct Extension Is Difficult | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.4 The Roles of the Different Set Domains | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Premeasures on Simple Sets | A | Preserve the existing definition fragment and domain/codomain; mark the omission in 2.2 with [Correction required]. |
| 2.1 Why Start with an Algebra? | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 Definition of a Premeasure | A | Preserve the existing definition fragment and domain/codomain; mark the omission in 2.2 with [Correction required]. |
| 2.3 Finite and Countable Additivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.4 Why the Union Must Remain in the Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. Outer Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 Definition of an Outer Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Construction by Countable Coverings | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Why the Infimum Appears | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.4 Proving That the Covering Construction Is an Outer Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.5 Why an Outer Measure Is Not Yet a Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. Carathéodory Measurability | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 The Splitting Criterion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 Why One Inequality Is Automatic | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Recovering Additivity Through Exact Splitting | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.4 Why the Criterion Is Not Ad Hoc | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. The Carathéodory Measurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.1 Closure Under Complements | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Closure Under Finite Unions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 Closure Under Countable Unions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.4 The Carathéodory Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.5 Completeness of the Restricted Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6. Recovering the Original Premeasure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.1 Agreement on the Original Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.2 Measurability of Sets in the Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.3 Inclusion of the Generated Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7. The Carathéodory Extension Theorem | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.1 Existence of an Extension | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.2 Comparison with Other Extensions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.3 Uniqueness Under Sigma-Finiteness | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.4 Where Sigma-Finiteness Is Used | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8. Comparing the Relevant Domains | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.1 The Original Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.2 The Generated Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.3 The Carathéodory Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.4 The Full Power Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.5 Completeness on the Different Domains | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 9. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 9.1 Why We Begin with Simple Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 9.2 Why We Pass Through an Outer Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 9.3 How Measurability Recovers Additivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 9.4 What the Extension Theorem Actually Accomplishes | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Introduction / Roadmap | C | An empty generic roadmap heading that duplicates the existing section order. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
| 2.2 / Def Premeasure label | C | Duplicates the immediately preceding Definition of a Premeasure heading. |
2026-07-26-folland-constructing-lebesgue-measure
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| From Interval Lengths to Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. Elementary Families of Intervals | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 Definition of an Elementary Family | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 Half-Open Intervals | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Why Half-Open Intervals Are Convenient | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.4 Finite Disjoint Unions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.5 Constructing the Interval Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Increasing Right-Continuous Functions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.1 Increasing Functions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 Right-Continuity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.3 Interval Increments | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.4 Unbounded Intervals | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.5 Why Right-Continuity Is Required | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. The Lebesgue–Stieltjes Premeasure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 Definition on Intervals | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Definition on Finite Disjoint Unions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Well-Definedness | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.4 Countable Additivity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.5 The Induced Premeasure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. The Lebesgue–Stieltjes Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 Extension to the Borel Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 Existence | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Uniqueness | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.4 Jumps and Point Masses | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.5 Continuous and Discrete Examples | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. Construction of Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.1 The Choice $F(x)=x$ | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Length on Half-Open Intervals | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 Borel Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.4 Lebesgue Outer Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.5 The Lebesgue Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6. Borel Measure and Lebesgue Completion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.1 Borel Measurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.2 Completion of Borel Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.3 Borel and Lebesgue Measurability | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.4 Why the Borel Measure Is Not Complete | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7. Structural Properties of Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.1 Translation Invariance | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.2 Scaling | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.3 Outer Regularity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.4 Inner Regularity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.5 Why Inner Regularity Uses Compact Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 8.1 From Intervals to a Complete Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.2 Why Topology Reappears Through Regularity | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.3 The Difference Between Borel and Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.4 How the Abstract Extension Theorem Becomes Concrete | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Introduction / Roadmap | C | An empty generic roadmap heading that duplicates the existing section order. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
2026-07-27-folland-product-measurable-spaces
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Why Product Measurable Spaces Are Needed | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. Coordinate Projections | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 Product Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 Coordinate Projections | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Preimages Under Coordinate Projections | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Product Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.1 Definition of the Product Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 The Universal Property | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.3 Why Coordinate Measurability Is Fundamental | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. Cylinder Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 One-Coordinate Cylinders | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Finite-Coordinate Cylinders | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Cylinder Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.4 Finite Versus Infinite Products | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. Coordinate Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 Generators of the Coordinate Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 Induced Generators of the Product Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Checking Product Measurability on Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. Measurable Rectangles | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.1 Definition of a Measurable Rectangle | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Rectangles as Intersections of Cylinders | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 Rectangular Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.4 Finite Products | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6. Product Borel Sigma-Algebras | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.1 The Basic Inclusion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.2 Equality Under Second Countability | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.3 Euclidean Product Spaces | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.4 Rectangular Borel Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.5 Countable Rectangular Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7. Infinite Products | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.1 Finite-Coordinate Dependence | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.2 Cylindrical Generators | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.3 Why Arbitrary Infinite Rectangles Are Different | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 8.1 Why Product Structures Are Defined Through Projections | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.2 Cylinders Versus Rectangles | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.3 Why Second Countability Appears | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.4 Preparing for Product Measures | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Introduction / Roadmap | C | An empty generic roadmap heading that duplicates the existing section order. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
2026-07-28-folland-cantor-set-and-completion
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Introduction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Why the Cantor Set Appears in Measure Theory | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1. Construction of the Cantor Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.1 The Iterative Construction | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.2 The Nested Closed Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 1.3 Closedness and Compactness | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2. Representations of Points in the Cantor Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.1 Ternary Expansions | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.2 Digits $0$ and $2$ | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 2.3 Binary Path Encoding | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3. Cardinality of the Cantor Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.1 Infinite Binary Sequences | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.2 Uncountability | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 3.3 Cardinality of the Continuum | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4. Measure of the Cantor Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.1 Length Removed at Each Stage | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.2 The Total Removed Length | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.3 Why the Cantor Set Has Measure Zero | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 4.4 Uncountable Sets of Measure Zero | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5. Subsets of Null Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.1 Cardinality of the Borel Sigma-Algebra | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.2 Cardinality of the Power Set of the Cantor Set | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 5.3 Existence of Non-Borel Subsets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6. Completion of Borel Lebesgue Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.1 Why Borel Lebesgue Measure Is Not Complete | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.2 Why Every Subset of the Cantor Set Is Lebesgue Measurable | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 6.3 The Lebesgue Sigma-Algebra as a Completion | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7. Comparing Classes of Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.1 Borel Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.2 Lebesgue Measurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.3 Nonmeasurable Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 7.4 The Main Inclusion Relations | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8. What I Finally Understood | A | Retain topic-specific reconstruction questions because they support review. |
| 8.1 Cardinality Does Not Determine Measure | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.2 Why Completion Adds New Sets | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| 8.3 Why Borel and Lebesgue Measurability Differ | A | Retain the existing topic and subheadings; leave the study content unwritten. |
| Summary | C | Empty and redundant with the topic-specific reconstruction section. |
2025-07-18-ieee754-errors-and-algorithms
| Original section / element | Grade | Assessment and edit |
|---|---|---|
| 1 Intro | B | The purpose of studying floating-point models and stable summation is needed. Condense the motivation to one sentence. |
| 1 Intro / standard history and rhetoric | C | Lists of manufacturers and transitions to the next section are unnecessary for reconstructing the later error analysis. Remove historical introductions and transitions without substantive content. |
| 1.1 What is IEEE 754? | A | Representation, rounding, and exceptions define the arithmetic model for the later algorithms. Retain the common normal-number formula and component table; flag missing normal, subnormal, and special cases. |
| 1.2 Float vs Double | B | Bit layouts and biases are essential, but the expanded formulas for the two formats repeat the same structure. Add biases to the table, remove duplicate expanded formulas, and request clarification of the float/double scope. |
| 2.1 Machine epsilon and relative, absolute error | A | These formulas provide a basis for comparing error magnitudes. Retain both error formulas and flag missing domain and epsilon conventions. |
| 2.2 Error types and exceptions | A | The distinctions among rounding, approximation, overflow, and underflow are worth reviewing. Preserve existing claims and flag classification errors; retain empty items as TODOs. |
| 2.2 unfinished transition | C | Unfinished phrases such as In short or It is because contain no argument. Remove incomplete filler and empty Proof labels. |
| 2.2 Absorption / loss of significance | A | Comparing a representative numerical failure with a stable rearrangement is a key example. Preserve the incorrect condition/displayed formula as an original claim, add revision guidance, and retain the code and output. |
| 2.3 Examples / non-associativity | A | This example shows how evaluation order changes the result. Preserve the short explanation and original Python code; remove the stray list marker. |
| 3 Kahan Two Sum | A | This study record examines a compensation algorithm and its error guarantee. Preserve the original code and error claim; flag naming, syntax, return-value, and error-model issues. |
| 3 Neumaier | A | The correction branch makes this a distinct technique rather than a duplicate Kahan example. Preserve the code and flag magnitude-comparison and correction-sign errors. |
| 3 Superaccumulator | A | Bit-level accumulation is a separate algorithmic idea. Preserve the code and flag issues with subnormal weights, exponent range, special values, and exactness. |
| 3 closing generalities about software / hardware | C | A conclusion with no specific next concept or argument. Remove it. |
2026-01-10-epsilon-optimal-cost-scaling-mcmf
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| 0 Caution | C | A promise of continued updates adds no mathematical content and duplicates status metadata. Remove it. |
| 1 Problem Definition | A | The objective and feasibility constraints determine the algorithm’s meaning. Explicitly retain the original incorrect objective and request corrections to the source, sink, lexicographic objective, capacities, and conservation conditions. |
| 2.1 Epsilon-Optimality / 2.2 Cost-Scaling | A | The scaling invariant and reduced costs are core structures. Preserve the existing expressions and flag notation conflicts and the missing residual-network definition. |
| 3 Push-Relabel Algorithm | B | The max-flow connection is useful, but describing the method merely as a heuristic cannot reconstruct correctness. Condense the introduction to one sentence and request distinctions between the invariants. |
| 3.1 Push all edges | A | Saturation at the start of a phase is a correctness dependency. Replace the empty Push statement with a TODO identifying the residual arcs and push amounts. |
| 3.2 Discharge / Relabel | A | This update connects preservation of epsilon-optimality with elimination of excess. Preserve the original potential update and flag its inconsistent reduced-cost convention. |
| 3.3 Exact solution / epsilon | A | The theorem connecting small epsilon to an exact optimum is essential. Do not supply the missing content; record proof tasks for integrality, thresholds, and feasible circulations. |
| 4 Pseudo Code and Implementation Tips | A | This is the original implementation to compare with the theory. Preserve the code and flag the epsilon loop, circulation reduction, cost reconstruction, and overflow. |
| 5 Time Complexity and Limitation | A | The costs and assumptions of the selected algorithm variant need review. Remove the unfinished sentence beginning Since and leave the required items as TODOs. |
| 6 Bipartite Matching | A | This application illustrates modeling with min-cost max-flow. Preserve the network model and flag issues involving cardinality priority and the additional arc. |
| 6 empty second application | C | No example or study content is present. Remove the empty numbered item. |
| 7 Reference | A | The existing max-flow reference must be distinguished from the direct cost-scaling source. Preserve the bibliography and add the cost-scaling authors’ source. |
2026-01-23-bertsimas-linear-optimization-ch1-1
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Chapter 1.1 / Motivation | A | Keep this unfinished outline without supplying its missing content. Retain headings and remove only parenthetical summary prompts. |
| Linear Optimization Problem | A | Variables, objective, and constraints are the main components of a formulation. Mark the existing items as TODOs. |
| Canonical Form | A | This is an existing mathematical statement. Preserve the equations and request dimensions, scalar fields, and form conventions. |
| Examples | A | The two models are distinct application headings. Keep Diet and Production planning as TODOs. |
| Remarks | A | Convexity and polyhedra connect to later duality and geometry. Preserve the TODOs without supplying new content. |
2026-02-04-steepest-descent-spd-convergence
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| 1 Motivation | B | The path from SD to CG/PCG and the comparison with direct solvers provide necessary motivation. Remove personal rhetoric and repetition, condensing the text to two sentences. |
| 2 Eigenvalue / Symmetric / Positive definite / SPD | A | These are essential hypotheses and terms for the convergence proof. Preserve the definitions and flag the incorrect positive-definiteness quantifier. |
| 2 Normal / Spectral Theorem | A | The eigenbasis proof technique and distinctions between scalar fields are important. Preserve the spectral theorem and basis-change reasoning; flag field, similarity, and adjoint errors. |
| 2 Q-transpose expanded row matrix | B | The projection-coefficient observation is needed, but displaying every row of the matrix is repetitive. Retain the row interpretation in one Korean sentence and remove the repeated matrix display. |
| 2 Rayleigh / Quadratic / strict convexity / Lipschitz / Condition number | A | These are prerequisites for line search and energy convergence. Preserve the definitions and flag vector dependence, the smoothness label, and norm assumptions. |
| 2 Iterative update / Line search | B | Directions and step sizes are needed, but much of the prose repeats the intuition. Preserve the update and argmin, condense the explanation, and leave a TODO distinguishing residuals from general directions. |
| 2 why argmin / intuition repetition | C | Merely restates the preceding argmin formula without adding reasoning. Remove it. |
| 3 Algorithm | A | This study code includes a residual update, periodic recomputation, and a stopping test. Preserve it and request clarification of residual terminology and cost comparisons. |
| 4 Existence and uniqueness | A | This argument uses SPD structure for invertibility, strict convexity, and coercivity. Preserve every proof dependency and request completion of the short gaps concerning the linear term and attainment of the minimum. |
| 4.a Constant step-size convergence | A | The error recurrence and spectral contraction are reusable proof techniques. Preserve the recurrence, eigenbasis bound, and stability-interval chain; flag the false equality and incorrect sufficient condition. |
| 4.a repeated inequality / update expansion | B | Repeated inequalities and arithmetic subtracting x-star are excessive. Retain the core error recurrence and interval implication, removing only redundant lines. |
| 4.b Energy norm / exact line search | A | This establishes what adaptive step sizes minimize. Preserve the existing norm and step formulas; flag the parameter, minimizer derivation, and zero-residual cases. |
| 4.b Rayleigh range / error expansion | B | The weighted-average observation and substitution r=-Ae are essential, but repeated expansions of matrix products can be condensed. Preserve the reason for the weighted average, spectral endpoints, cross term, and energy identity; remove routine intermediate distributive steps. |
| 4.b Spectral moments / Kantorovich-type inequality | A | Although long, this proof explains the main theorem’s technique and assumptions. Preserve the three moments, choices of a_i/b_i, coordinatewise inequality, AM–GM, and final factor; flag square, coefficient, implication, and zero-coordinate errors. |
| 4 Speed of convergence | B | The global bound and tolerance connection are needed, but induction is written twice for the same expression. Preserve the q^k estimate and iteration-count formula, remove the repeated induction display, and flag endpoint and bound-interpretation issues. |
| 4 Python / MATLAB validation | A | This is a study record comparing theory and experiment. Preserve all code and existing output; flag the function-output mismatch and limits of experimental interpretation. |
| 5 Conditioning and cost | B | The connection between conditioning and sparse-solver cost is essential but repeats the earlier bound discussion. Retain formulas and limitations, condense lengthy prose, and flag reduction and complexity issues. |
| 5 Zig-zag / next direction | A | Residual orthogonality and eigenbasis updates are key distinctions leading to CG. Preserve both observations; flag the componentwise-decrease inference and the comparison with the CG bound. |
| 6 Reference | A | These links support the original reading and follow-up verification. Preserve every existing URL and add the relevant sections of Saad’s author-hosted book. |
2026-02-09-conjugate-gradient
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| 1 Motivation | B | The limitations of SD and reuse of previous residual information are the central motivation. Condense rhetoric and repetition, and flag the confusion with quadratic convergence. |
| 1 Cyclic subspace / residual projection | A | This is a key conceptual connection to Krylov spaces. Preserve the cyclic span and original orthogonality claim; flag map/matrix identification and the invalid implication. |
| 2 Prerequisite | A | Krylov spaces, invariant subspaces, projection, and Petrov–Galerkin are the required major concepts. Standardize empty definition labels as TODOs and flag missing SPD/A-inner-product prerequisites. |
| 2 Wolfe condition & condition | C | An unexplained, unwritten placeholder outside the current proof dependencies for linear SPD CG. Remove it. |
| 3 Algorithm & Code | A | This original implementation allows comparison of the recurrence with the theory. Preserve the MATLAB code and add tasks explaining finite precision and termination status. |
| 4 Correctness | A | This is the most important current proof gap. Preserve the Chebyshev/ratio draft as an original claim and record tasks for the correctness chain and precise bound statement. |
| 5 Improvements but limitations | B | The geometric motivation leading to preconditioning is needed. Remove two empty fragments, retain a one-sentence ellipsoid motivation, and request equivalence/SPD details. |
| 6 Reference | A | These are existing study references. Preserve the bibliography and link Saad’s author-hosted PDF. |
2026-07-31-measure-theory-seminar-03
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Today’s Roadmap | C | Remove roadmap headings/lists that duplicate the contents; retain the substantive topic sections. |
| Proposition 2.1 | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Proposition 2.3 | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Proposition 2.13 | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Monotone Convergence Theorem | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Questions | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
2026-08-07-fatou-dct
| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Today’s Roadmap | C | Remove roadmap headings/lists that duplicate the contents; retain the substantive topic sections. |
| Fatou’s Lemma | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Statement | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Proof | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Intuition | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Remarks | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Dominated Convergence Theorem | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Statement | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Proof | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Step 1: Apply Fatou to $g+f_n$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Step 2: Apply Fatou to $g-f_n$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Why Domination Is Necessary | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| Comparison with MCT and Fatou’s Lemma | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Monotone Convergence Theorem | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Fatou’s Lemma | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Dominated Convergence Theorem | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
| Examples | C | Remove the routine example repeating direct integration of x/n and application of DCT; preserve the concentration counterexample. |
| Summary | B | Consolidate repeated proof explanations and theorem restatements into proof ideas and an assumptions/conclusions table. |
2026-08-14-Lp-space-and-inequalities-04
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| Original section / element | Grade | Assessment and edit |
|---|---|---|
| Roadmap | C | Remove roadmap headings/lists that duplicate the contents; retain the substantive topic sections. |
| 1. From Lebesgue Integration to $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 1.1 What Did $L^1$ Measure? | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 1.2 Why Consider $|f|^p$? | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 1.3 How Does $p$ Change the Notion of Size? | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 1.4 From Integrability to Function Spaces | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2. $L^p$ Spaces | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.1 Definition for $1 \le p < \infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.2 The Case $p=\infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.3 Essential Supremum | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.4 Equality Almost Everywhere | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.5 Why We Need Equivalence Classes | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.6 The Space $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.7 Basic Properties of the $L^p$-Norm | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 2.8 First Examples: $L^1$, $L^2$, and $L^\infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3. Young’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3.1 Conjugate Exponents | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3.2 Statement of Young’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3.3 Proof | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3.4 Equality Case | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 3.5 Why Young’s Inequality Matters | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 4. Hölder’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.1 Statement | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.2 The Normalization Idea | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.3 Proof Using Young’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.4 Endpoint Case: $p=1,\ q=\infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.5 Cauchy–Schwarz as the Case $p=q=2$ | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 4.6 Product Integrability | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 4.7 Equality Case | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 4.8 Why Hölder Is Important | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 5. Minkowski’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.1 Statement | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.2 The Easy Case $p=1$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.3 The Case $1<p<\infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.4 Applying Hölder’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.5 Why the Conjugate Exponent Appears | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 5.6 Completion of the Proof | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 5.7 Minkowski as the Triangle Inequality | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 5.8 Conclusion: $L^p$ Is a Normed Vector Space | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 6. Relations Between $L^p$ Spaces | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 6.1 Does Larger $p$ Mean a Smaller Space? | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 6.2 The Finite Measure Case | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 6.3 The Inclusion $L^q(X)\subset L^p(X)$ | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 6.4 Proof Using Hölder’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 6.5 Norm Estimate | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 6.6 Why the Finite Measure Assumption Matters | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 6.7 What Happens on Infinite Measure Spaces? | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 7. Examples and Counterexamples | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.1 Power Functions Near Zero | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.2 Power Functions Near Infinity | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.3 A Function in $L^p$ but Not in $L^q$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.4 A Function in $L^q$ but Not in $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.5 Why There Is No Universal Inclusion on $\mathbb{R}$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 7.6 Comparing $L^1$, $L^2$, and $L^\infty$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8. Completeness of $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.1 Normed Spaces and Cauchy Sequences | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.2 Complete Normed Spaces | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.3 Completeness Theorem for $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.4 Start with an $L^p$-Cauchy Sequence | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.5 Choosing a Rapidly Convergent Subsequence | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.6 Constructing an Auxiliary Sequence | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.7 Control Using Minkowski’s Inequality | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.8 Passing to the Limit | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.9 Using the Previous Convergence Theorems | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.10 Pointwise Almost Everywhere Convergence | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.11 Convergence in $L^p$ | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.12 Returning to the Original Cauchy Sequence | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 8.13 Conclusion | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 9. Why $L^2$ Is Special | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 9.1 Inner Product Spaces | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 9.2 The $L^2$ Inner Product | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 9.3 The Norm Induced by the Inner Product | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 9.4 Hilbert Spaces | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 9.5 Banach vs. Hilbert | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 9.6 Why $L^2$ Will Matter Later | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 10. Where $L^p$ Goes Next | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 10.1 Measuring Functions | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 10.2 What If We Also Want to Measure Derivatives? | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 10.3 Weak Derivatives | B | Preserve content and dependencies while merging closely related headings or short explanations into adjacent sections. |
| 10.4 Preview of Sobolev Spaces | A | Preserve the key definition, theorem, proof step, assumption, connection, or TODO for the author to complete. |
| 10.5 Next Seminar | B | Replace the generic functional-analysis list with links to Week 05 duality and its prerequisites, matching the next point in the requested study sequence. |
| Summary | B | Remove repeated inequalities and Banach/Hilbert conclusions, retaining only the central dependency chain. |
Archive / Navigation coverage
| Location | Grade | Assessment |
|---|---|---|
| Archive seminar 01–02 | C / Original preserved | Files are empty or contain only front matter. Do not count them as completed study. |
| Archive weak derivatives | A / Original preserved | Record the assumptions and notation issues in the single fragment in the review report. |
| Archive originals / MCMF backups | C / Original preserved | Duplicate copies of the active note. Review the active note without modifying the historical snapshots. |
| 4 project pages | A | Preserve the distinction between current evidence and planned work, and add study-prerequisite links. |
| Home / Notes / Measure Theory / Start Here / tabs and indexes | A | Preserve navigation and research directions and link the new roadmap. There is no mathematical body text. |
| Templates / maintenance docs | A | Retain as authoring/maintenance material. Do not include it in the count of actual study records. |