Potted Measure Theory

notes
llm
mathematics
probability
A quick summary of measure theory on the real line.
Author

Stephen J. Mildenhall

Published

2024-05-22

Introduction

This post is another comparison of LLMs. I asked for a quick summary of measure theory on the real line. I had some follow up questions for GPT shown after the comparison—I was using these notes to refresh my memory!

Question

Please write me a short summary of measure theory. Start with the length of an interval defining length on Borel sets. The outer measure (and inner) and hence to measurable sets and completions, and Lebesgue measurable sets. Include relevant definitions, the key results, and any interesting examples.

Responses

GPT 4.o

Summary of Measure Theory

Length of an Interval

The length of an interval \([a, b] \subset \mathbb{R}\) is defined as \(b - a\). This concept extends to more complex sets in \(\mathbb{R}\) through measure theory.

Borel Sets

Borel sets are the \(\sigma\)-algebra generated by open intervals in \(\mathbb{R}\). This is the smallest \(\sigma\)-algebra containing all open intervals.

Outer Measure

The outer measure of a set \(A \subset \mathbb{R}\) is defined as: \[ m^*(A) = \inf \left\{ \sum_{k=1}^{\infty} \ell(I_k) : A \subseteq \bigcup_{k=1}^{\infty} I_k \right\} \] where \(I_k\) are intervals and \(\ell(I_k)\) denotes their length.

Inner Measure

The inner measure is defined similarly but considers coverings by complements of sets: \[ m_*(A) = \sup \left\{ \sum_{k=1}^{\infty} \ell(I_k) : \bigcup_{k=1}^{\infty} I_k \subseteq A \right\}. \]

Measurable Sets

A set \(A \subseteq \mathbb{R}\) is measurable if for every \(\epsilon > 0\), there exists an open set \(O\) such that \(A \subseteq O\) and \(m^*(O \setminus A) < \epsilon\).

Lebesgue Measure

Lebesgue measure extends the concept of length to more complex sets and is defined using outer measure:

  1. A set \(A\) is Lebesgue measurable if for any set \(E\), \[ m^*(E) = m^*(E \cap A) + m^*(E \cap A^c). \]
  2. The Lebesgue measure \(m\) of a set \(A\) is \(m(A) = m^*(A)\).

Key Results

  1. Carathéodory’s Criterion: A set \(A \subseteq \mathbb{R}\) is Lebesgue measurable if and only if for every set \(E \subseteq \mathbb{R}\), \[ m^*(E) = m^*(E \cap A) + m^*(E \cap A^c). \]
  2. Countable Additivity: If \(\{A_i\}\) is a countable collection of disjoint Lebesgue measurable sets, \[ m\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} m(A_i). \]
  3. Completion of Measure: The Lebesgue measure can be extended to include all subsets of sets of measure zero, forming the completed measure.

Examples

  1. Non-Measurable Set: Vitali sets are examples of sets that are not Lebesgue measurable. They are constructed using the axiom of choice and cannot be assigned a consistent measure.
  2. Cantor Set: The Cantor set is a classic example of a set with Lebesgue measure zero but uncountable cardinality.

Conclusion

Measure theory generalizes the concept of length from intervals to more complex sets, providing a rigorous foundation for integration and probability. Lebesgue measure plays a crucial role in modern analysis, enabling the precise treatment of functions and sets beyond simple geometrical objects.

Claude 3

Measure theory is a branch of mathematics that provides a rigorous foundation for the concept of length, area, and volume, and generalizes these notions to more abstract spaces. It is a key tool in various fields, including probability theory, functional analysis, and mathematical physics.

  1. Length of an interval: The length of an interval [a, b] is defined as b - a. This notion of length is extended to define a measure on the Borel sets.

  2. Borel sets: The Borel σ-algebra on a topological space X is the smallest σ-algebra containing all open sets. Elements of the Borel σ-algebra are called Borel sets.

  3. Outer measure: For any set A, the outer measure μ*(A) is defined as the infimum of the sum of lengths of countable collections of open intervals that cover A. The outer measure satisfies countable subadditivity: μ(∪ᵢAᵢ) ≤ ∑ᵢμ(Aᵢ) for countable collections {Aᵢ}.

  4. Measurable sets: A set A is measurable if for every set E, μ*(E) = μ*(E ∩ A) + μ*(E ∩ Aᶜ). In other words, A is measurable if it splits every set into two parts in a way that is consistent with the outer measure.

  5. Lebesgue measure: The Lebesgue measure is the unique complete measure on the Borel σ-algebra that coincides with the outer measure on Borel sets. A set is Lebesgue measurable if it is measurable with respect to the Lebesgue measure.

  6. Completions: The completion of a measure space (X, Σ, μ) is the measure space (X, Σ’, μ’) where Σ’ consists of all sets of the form A ∪ B, where A ∈ Σ and B is contained in a null set (a set of measure zero).

Key results:

  • The Lebesgue measure extends the notion of length to a larger class of sets than the Borel sets.
  • The Lebesgue measure is translation invariant: λ(A + x) = λ(A) for all measurable sets A and real numbers x.
  • The Lebesgue measure is complete: every subset of a null set is measurable.

Interesting examples:

  • The Cantor set is an example of an uncountable set with Lebesgue measure zero.
  • The Vitali set is an example of a non-measurable set (assuming the Axiom of Choice).
  • The Borel-Cantelli lemma uses measure theory to prove that, under certain conditions, a sequence of events will occur infinitely often with probability 1.

Measure theory provides a solid foundation for integration theory, allowing the definition of the Lebesgue integral, which extends the Riemann integral to a larger class of functions. This, in turn, is crucial for the development of functional analysis and its applications in various fields of mathematics and physics.

Gemini Advanced

Absolutely! Here’s a concise summary of measure theory, touching on the key points you requested:

Measure Theory: A Framework for Measuring Sets

Measure theory generalizes our intuitive notions of length, area, and volume to a wider class of sets.

The Beginning: Length of Intervals

  • We start with the real line. The length of an interval (a, b) is simply b - a.
  • This is a measure on intervals: a function that assigns a non-negative number to each interval, and is additive (the length of two disjoint intervals is the sum of their lengths).

Borel Sets and Lebesgue Measure

  • Borel sets: We want to measure more than just intervals. Borel sets are formed by taking countable unions, intersections, and complements of intervals. This collection is rich enough to include most sets encountered in analysis.
  • Lebesgue measure: We extend the idea of interval length to Borel sets. Lebesgue measure assigns a “length” to each Borel set, preserving the properties of being non-negative and countably additive (the measure of a countable union of disjoint Borel sets is the sum of their measures).

Outer Measure and Measurable Sets

  • Outer measure: To measure even more sets, we introduce outer measure. It assigns a “size” to any subset of the real line, not just Borel sets.
  • Measurable sets: However, outer measure isn’t always well-behaved. We define a set to be measurable if its outer measure “agrees” with the Lebesgue measure of Borel sets it interacts with.
  • Carathéodory’s Theorem: This important result states that measurable sets form a sigma-algebra (a collection closed under complements and countable unions), and Lebesgue measure is countably additive on measurable sets.

Completeness and Lebesgue Measurable Sets

  • Complete measure space: A measure space is complete if every subset of a null set (a set of measure zero) is also measurable.
  • Lebesgue measurable sets: We complete the Lebesgue measure space by adding all subsets of null sets. This results in the collection of Lebesgue measurable sets.

Key Results and Examples

  • Vitali Set: A famous example of a non-measurable set. Its existence demonstrates the necessity of restricting ourselves to measurable sets when dealing with measure theory.
  • Cantor Set: A fascinating example of an uncountable set with Lebesgue measure zero. It’s constructed by repeatedly removing the middle thirds of intervals.

Applications

Measure theory underpins modern probability theory, integration theory (Lebesgue integration), and has applications in areas such as functional analysis, ergodic theory, and even mathematical finance.

Let me know if you’d like a deeper dive into any of these topics!

Followup Questions (GPT only)

what is the relation between inner and outer measure?

The inner and outer measures are closely related concepts in measure theory, both aiming to quantify the “size” of a set, but they approach this task from opposite directions.

Relationship Between Inner and Outer Measure

  1. Definitions:
    • Outer Measure (\(m^*\)): The outer measure of a set \(A \subseteq \mathbb{R}\) is defined as the infimum of the lengths of countable collections of intervals that cover \(A\). \[ m^*(A) = \inf \left\{ \sum_{k=1}^{\infty} \ell(I_k) : A \subseteq \bigcup_{k=1}^{\infty} I_k \right\} \]
    • Inner Measure (\(m_*\)): The inner measure of a set \(A \subseteq \mathbb{R}\) is defined as the supremum of the lengths of countable collections of intervals contained within \(A\). \[ m_*(A) = \sup \left\{ \sum_{k=1}^{\infty} \ell(I_k) : \bigcup_{k=1}^{\infty} I_k \subseteq A \right\} \]
  2. Bounds on the Measure of a Set:
    • For any set \(A \subseteq \mathbb{R}\), the inner measure is always less than or equal to the outer measure: \[ m_*(A) \leq m^*(A) \]
    • This inequality holds because the intervals used in the definition of the inner measure are a subset of those used in the definition of the outer measure.
  3. Measurable Sets:
    • A set \(A\) is measurable if its outer measure equals its inner measure: \[ m^*(A) = m_*(A) \]
    • For Lebesgue measurable sets, this equality implies that the measure \(m(A)\) is well-defined and consistent.
  4. Examples:
    • Closed Interval \([a, b]\): For a closed interval, the inner measure and outer measure are both \(b - a\), making it measurable.
    • Non-measurable Set: For some sets (like Vitali sets), the inner and outer measures can differ, indicating that they are not Lebesgue measurable.

Conclusion

The inner and outer measures provide a way to approximate the size of a set from the inside and outside, respectively. Their equality characterizes measurable sets, a cornerstone of measure theory and the foundation for defining the Lebesgue measure.

how hard is it to prove that m* is a measure on the set of measurable sets as defined above?

Proving that \(m^*\) (the outer measure) restricted to the set of measurable sets \(\mathcal{M}\) is a measure involves several steps. Here’s a brief outline of the proof, emphasizing the key ideas and the level of difficulty:

Steps to Prove \(m^*\) is a Measure

  1. Show \(\mathcal{M}\) is a \(\sigma\)-algebra:
    • Closed under Complement: If \(A \in \mathcal{M}\), then \(A^c \in \mathcal{M}\).
    • Closed under Countable Unions: If \(\{A_i\} \subseteq \mathcal{M}\), then \(\bigcup_{i=1}^{\infty} A_i \in \mathcal{M}\).
  2. Countable Additivity on Disjoint Sets:
    • If \(\{A_i\} \subseteq \mathcal{M}\) are disjoint, then \[ m^*\left( \bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty} m^*(A_i) \]

Detailed Steps

  1. Prove \(\mathcal{M}\) is a \(\sigma\)-algebra:
    • Closed under Complement:
      • Given \(A \in \mathcal{M}\), we need to show \(A^c \in \mathcal{M}\).
      • By definition, \(A\) is measurable if for all \(E \subseteq \mathbb{R}\): \[ m^*(E) = m^*(E \cap A) + m^*(E \cap A^c) \]
      • Using this property, it follows that \(A^c\) satisfies the same condition.
    • Closed under Countable Unions:
      • Let \(\{A_i\} \subseteq \mathcal{M}\).
      • We need to show \(\bigcup_{i=1}^{\infty} A_i \in \mathcal{M}\).
      • Using the measurability condition and properties of \(m^*\), we can prove that for any \(E \subseteq \mathbb{R}\): \[ m^*(E) = m^*\left( E \cap \bigcup_{i=1}^{\infty} A_i \right) + m^*\left( E \cap \left( \bigcup_{i=1}^{\infty} A_i \right)^c \right) \]
  2. Prove Countable Additivity:
    • Disjoint Sets:
      • Suppose \(\{A_i\} \subseteq \mathcal{M}\) are disjoint.
      • We need to show: \[ m^*\left( \bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty} m^*(A_i) \]
      • This follows from the subadditivity of \(m^*\) and the fact that the sets are disjoint: \[ m^*\left( \bigcup_{i=1}^{\infty} A_i \right) \leq \sum_{i=1}^{\infty} m^*(A_i) \]
      • For the reverse inequality, use the definition of \(m^*\) and properties of disjoint sets.

Difficulty

The difficulty of this proof lies in the careful handling of the properties of outer measure \(m^*\) and the definition of measurability. The key challenges include:

  • Demonstrating closure properties of \(\mathcal{M}\).
  • Utilizing the definition of outer measure to establish countable additivity, particularly for disjoint sets.

These proofs are intricate but manageable with a solid understanding of measure theory fundamentals and properties of \(m^*\). They typically form part of an advanced undergraduate or graduate course in real analysis or measure theory. The proofs require a rigorous approach to epsilon-delta arguments and careful manipulation of sets and their measures.