Measure Theory
Table of Contents
1. Measure Space
- \((X, \Sigma, \mu)\)
1.1. Definition
- Triple of
- a set \(X\),
- σ-Algebra \(\Sigma\) over \(X\),
- a measure \(\mu\) on \(\Sigma\).
1.2. σ-Algebra
- σ-Field
1.2.1. Definition
- σ-algebra \(\Sigma\) is a subset of the power set of a set \(X\),
such that:
- \(X \in \Sigma\)
- Closed under complementation: \(A \in \Sigma \implies X\backslash A \in \Sigma\)
- Closed under countable unions: \(A_1, A_2, \dots \in \Sigma \implies \bigcup_{i\in \mathbb{N}} A_i \in \Sigma\)
1.2.2. Properties
- It immediately follows from the definition that:
- \(\varnothing \in\Sigma\)
- Closed under countable intersections by the De Morgan's laws
1.2.3. Separable σ-Algebra
- σ-Algebra that is also a separable space with metric \(\rho(A, B) = \mu(A\vartriangle B)\), where \(\mu\) is a finite measure, and \(\vartriangle\) is the symmetric difference.
1.2.4. Generation
- \(\sigma\)-Algebra generated by a single subset: \(\sigma(A)\) where \(A\subset X\).
- \(\sigma\)-Algebra generated by a function: \(\sigma(f)\) for a measurable function \(f: X \to Y\).
1.3. Measure
- Measure \(\mu\) is a set function from
\(\Sigma\) to the extended real number line with:
- \(\mu(\varnothing) = 0\)
- Non-Negativity: \(\forall E\in \Sigma, \mu(E) \ge 0\)
- Countable Additivity (or \(\sigma\)-additivity): \(\forall \{E_k\}_{k\in \mathbb{N}}\subset \Sigma\) of pairwise disjoint sets, \[ \mu\left(\bigcup_{k\in \mathbb{N}} E_k\right) = \sum_{k\in \mathbb{N}}\mu(E_k). \]
1.3.1. Signed Measure
- If the non-negativity condition is omitted, the measure becomes a signed measure.
1.4. Measurable Space
- The pair \((X, \Sigma)\) of a set and its σ-Algebra , is a measurable space.
1.4.1. Measurable Set
- The members of \(\Sigma\) are measurable sets.
2. Non-Measurable Set
2.1. Vitali Set
Using the axiom of choice, form a set \( V \) from \( \mathbb{R}/\mathbb{Q} \) by choosing one element in each equivalence class.
This set is not Lebesgue measurable.
Notice that \[ [0,1] \subseteq \bigcup_{q \in \mathbb{Q}} \{ v + q : v\in V\} \subseteq [-1,2] \] and by the axiom of measure \[ \mu([0,1]) = 1 \le \sum_{\text{countable infinity}} \mu(V) \le \mu([-1,2]) = 3. \] Such a number \( \mu(V) \) does not exists, and \( V \) is not Lebesgue measuable.
3. Measurable Function
- Function between two measurable spaces that preserves the structure.
3.1. Definition
- For measurable spaces \((X, \Sigma)\) and \((Y, \mathrm{T})\), a measurable function \(f: X\to Y\) satisfies: \[ \forall E\in T, f^{-1}[E] \in \Sigma. \]
4. Radon-Nikodym Derivative
4.1. Radon-Nikodym Theorem
- For two σ-finite measures \(\mu\) and \(\nu\) on a measure space \((X, \Sigma)\), if \(\nu \ll \mu\), then there exists a Σ-measurable function \(f\colon X\to [0, \infty)\), such that for any measurable set \(A\in \Sigma\): \[ \nu(A) = \int_A f\,d\mu. \]
- \(\nu\) is absolutely continuous with respect to \(\mu\).
4.2. Definition
- The function \(f\) is uniquely defined up to a \(\mu\)-null set, and is defined to be the Radon-Nikodym derivative \(f\), which is commonly written as \[ \frac{d\nu}{d\mu}. \]
5. Borel Measure
5.1. Definition
- Any measure defined on the Borel (σ-)algebra—the smallest σ-algebra containing all open sets—of a topological space.
5.2. Borel Functor
- \[ \mathbf{Bor}: \mathbf{Top}_{2CHaus} \to \mathbf{Meas} \]
- It preserves finite products: \(\mathbf{Bor}(X\times Y) \cong \mathbf{Bor}(X)\times \mathbf{Bor}(Y)\).
6. Lebesgue Measure
- The measure of the Euclidean space.
6.1. Definition
- The Lebesgue outer measure is: \[ \lambda^*(E) = \inf\left\{\sum_{k=1}^\infty \ell(I_k): \{I_k\}_{k\in \mathbb{N}}\text{ is a sequence of open intervals with }E\subset \bigcup_{k=1}^\infty I_k\right\} \] where \(\ell(I_k)\) is the length of the interval.
6.2. Lebesgue Measurable
6.2.1. Carathéodory Criterion
- \(E\) satisfies the criterion if: \[ \forall A\subseteq \mathbb{R}, \lambda^*(A) = \lambda^*(A\cap E) +\lambda^*(A\cap E^C). \]
- The sets \(E\) that satisfy the Carathéodory criterion forms a σ-Algebra, and they are Lebesgue-measurable, with the Lebesgue measure \(\lambda(E) = \lambda^*(E)\).
6.2.2. Non-Example
- Vitali Sets