# Measure Theory

## 1. Supremum, Infimum, Limit Superior, and Limit Inferior

- Upper and Lower Bounds of Subsets of Real Numbers
- The Supremum and Infimum of a Nonempty Subset of Real Numbers
- Properties of the Supremum and Infimum of a Nonempty Subset of Real Numbers
- The Supremum and Infimum of the Sum of Nonempty Subsets of Real Numbers
- The Supremum and Infimum Properties for Subsets of Sets of Real Numbers
- The Limit Superior and Limit Inferior of Sequences of Real Numbers
- The Limit Superior and Limit Inferior of Functions of Real Numbers
- Topologies on Sets

## 2. Algebras of Sets

## 3. The Lebesgue Measure and Lebesgue Measurable Sets

###### 3.1. The Lebesgue Outer Measure

- Set Functions ( Examples 1 )
- The Length Function on the Set of Intervals
- The Lebesgue Outer Measure
- The Lebesgue Outer Measure of Intervals
- Countable Subadditivity of the Lebesgue Outer Measure
- Translation Invariance of the Lebesgue Outer Measure
- The Lebesgue Outer Measure of Open Sets and G𝛿 Sets Containing a Set
- Example Problems Regarding the Lebesgue Outer Measure
- The Lebesgue Outer Measure Review

###### 3.2. Lebesgue Measurable Sets

- Lebesgue Measurable Sets
- A Property of Finite Mutually Disjoint Collections of Lebesgue Measurable Sets
- The Union of a Finite Collection of Lebesgue Measurable Sets is Lebesgue Measurable
- The Union of a Countable Collection of Lebesgue Measurable Sets is Lebesgue Measurable
- The Lebesgue Measurability of Intervals
- The Lebesgue Measurability of Translates of Lebesgue Measurable Sets
- The Regularity Properties of the Lebesgue Measure
- The Collection of Lebesgue Measurable Sets
- The Existence of a Non-Lebesgue Measurable Set
- Example Problems Regarding Lebesgue Measurable Sets 1
- Example Problems Regarding Lebesgue Measurable Sets 2
- Lebesgue Measurable Sets Review

###### 3.3. The Lebesgue Measure

## 4. Lebesgue Measurable Functions

###### 4.1. Lebesgue Measurable Functions

## 5. The Lebesgue Integral

###### 5.1. The Lebesgue Integral of Simple Functions

- The Lebesgue Integral of Simple Functions
- The Linearity Property of the Lebesgue Integral of Simple Functions
- The Monotonicity Property of the Lebesgue Integral of Simple Functions
- The Lebesgue Integral of a Simple Function Defined on a Union of Disjoint Sets

###### 5.2. The Lebesgue Integral of Bounded Lebesgue Measurable Functions

- The Lebesgue Integral of Bounded Functions
- The Lebesgue Integral of Bounded, Riemann Integrable Functions
- The Lebesgue Integral of Bounded, Lebesgue Measurable Functions
- The Linearity Property of the Lebesgue Integral of Bounded, Lebesgue Measurable Functions
- The Monotonicity Property of the Lebesgue Integral of Bounded, Lebesgue Measurable Functions

- The Pointwise Bounded Convergence Theorem for Uniformly Convergent Sequences of Functions
- The Uniform Bounded Convergence Theorem for Pointwise Convergent Sequences of Functions

###### 5.3. The Lebesgue Integral of Nonnegative Lebesgue Measurable Functions

- The Lebesgue Integral of Nonnegative Lebesgue Measurable Functions
- The Linearity Property of the Lebesgue Integral of Nonnegative Lebesgue Measurable Functions
- Fatou's Lemma for Nonnegative Lebesgue Measurable Functions

###### 5.4. The General Lebesgue Integral

- The Lebesgue Integral for Lebesgue Measurable Functions
- The Linearity Property of the Lebesgue Integral of Lebesgue Integrable Functions
- The Monotonicity Property of the Lebesgue Integral of Lebesgue Integrable Functions
- The Comparison Test for Lebesgue Integrability
- Additivity over Domains of Integration of the Lebesgue Integral of Lebesgue Integrable Functions

## 6. Differentiation and the Lebesgue Integral

###### 6.1. Vitali Covers

- Vitali Covers of a Set
- The Vitali Covering Lemma Part 1
- The Vitali Covering Lemma Part 2
- Vitali Cover Review

###### 6.2. Lebesgue's Theorem for Differentiability of Monotone Functions

- Upper and Lower Derivatives of Real-Valued Functions
- Lebesgue's Theorem for the Differentiability of Monotone Functions
- The Lebesgue Integral of the Derivative of an Increasing Function
- Lebesgue's Theorem and Differentiability of Monotone Functions Review

###### 6.3. Functions of Bounded Variation and Absolutely Continuous Functions

- Functions of Bounded Variation on Closed Intervals
- Absolute Continuity
- Absolutely Continuous Functions are of Bounded Variation
- Functions of Lebesgue Integrals
- Integral Criteria for Functions to be Zero Almost Everywhere
- The Derivative of Functions of Lebesgue Integrals
- Classification of Absolutely Continuous Functions
- Absolutely Continuous Functions Review

## 7. General Measure Spaces

###### 7.1. General Measurable Spaces and Measure Spaces

- General Measurable Spaces and Measure Spaces
- The Counting Measure
- The Dirac Measure at x
- Basic Properties of Measure Spaces
- Complete Measure Spaces
- The Completion of a Measure Space
- The Borel-Cantelli Lemma

###### 7.2. General Measurable Functions

- General Measurable Functions
- Sums, Multiples, and Products of Measurable Functions
- The Simple Function Approximation Lemma and Theorem for General Measurable Spaces

###### 7.3. Abstract Integration over General Measure Spaces

- The Integral of Nonnegative Measurable Functions
- Chebyshev's Inequality for Nonnegative Measurable Functions
- Fatou's Lemma for Nonnegative Measurable Functions
- The Monotone Convergence Theorem for Nonnegative Measurable Functions
- Beppo Levi's Lemma for Nonnegative Increasing Measurable Functions
- The Linearity Property of the Integral of Nonnegative Measurable Functions
- The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions
- Integrals of Nonnegative Measurable Functions that Equal Zero

- The Integral of Measurable Functions
- The Comparison Test for Integrability
- The Dominated Convergence Theorem for Measurable Functions

###### 7.4. Outer Measures on Measurable Spaces

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###### References

- 1. Real Analysis (4th Edition) by Halsey Royden and Patrick Fitzpatrick.