2 Measures and σ-algebra

This section develops the abstract machinery of measures and \(\sigma\)-algebras:

  1. Definitions of \(\sigma\)-algebra, product \(\sigma\)-algebra, and monotone class theorem.
  2. Measure spaces: \(\sigma\)-finite measures, measures of nested intersections or unions; sufficient conditions for measurable functions; uniqueness of \(\sigma\)-finite measures (a prime demonstration of the monotone class theorem).
  3. Measurable functions: composition establishing measurability using the generators of the \(\sigma\)-algebra.
  4. Carathéodory theorem: constructing a measure by restricting an outer measure.

Key takeaways:

  1. A measure space specifies the well-behaved, measurable sets (\(\sigma\)-algebra), together with a function (called measure) which assigns a measure value to each measurable set. The key desired property of a measure is countable additivity under disjoint unions.
  2. Monotone class theorem 2.1: the monotone class freely generated by an algebra of sets is also its freely-generated \(\sigma\)-algebra.
  3. The product algebra \(\Sigma_1\times \Sigma_2\) is universal and satisfies the section property (proposition ??).
  4. The preimage-operation commutes with all closure operations of the \(\sigma\)-algebra, the key properties of which is countability; to check measurability of a function it suffices to check the generators of the codomain \(\sigma\)-algebra (proposition 2.6).
  5. General method of constructing measures: restricted finite additive measure \(\to\) unrestricted countably subadditive (outer) measure \(\to\) restricted countably additive measure.
  6. Measurable sets by the Carathéodory criteria (definition 2.10) are closed under \(\sigma\)-algebra operations.

To motivate the careful construction of a rigorous measure:

  • We need more than finite additivity in order to exchange integrals and limits.
  • Uncountable additivity is too strong: points have measure zero, but intervals should have nontrivial measure.
  • There exists subsets of \(\mathbb R^n\) which are too irregular to accomodate additivity, so we need the machinery of \(\sigma\)-algebra to describe the measurable sets.

σ-algebra

Definition 2.1 (algebra of sets) A collection \(\mathcal A\) of sets is an algebra of sets if it is closed under finite union and differences: \[ A, B\in \mathcal A\implies A\cup B, A-B, B-A\in \mathcal A \]

Definition 2.2 (σ-algebra) Given a set \(\Omega\), \(\Sigma\subset 2^\Omega\) is a \(\sigma\)-algebra on \(\Omega\) if:

  1. \(\emptyset\in \Sigma\).
  2. Closure under complement: \(E\in \Sigma\implies \Omega-E\in \Sigma\).
  3. Closure under countable union: \(\{E_j\}\in \Sigma\implies \bigcup_{k=1}^\infty E_k \in \Sigma\).

Proposition 2.1 (derived properties of σ-algebras) Given a \(\sigma\)-algebra \(\Sigma\) on \(\Omega\), then

  1. \(\Omega\in \Sigma\).
  2. Closure under finite union, intersection, and difference: \[ D, E\in \Sigma\implies D\cup E, D\cap E, D-E\in \Sigma \]
  3. \(E_1, \cdots\in \Sigma\implies \bigcap_{k=1}^\infty E_k \in \Sigma\)

Proof: (1) follows from \(\Omega=\Omega-\emptyset\). (2) follows from \(D\cap E = (\Omega-D)\cup (\Omega_E)\) and \(D-E=(D\cap E)\cup E\). (3) follows from \(\bigcap_{k=1}^\infty E_k = \Omega - \bigcup_{k=1}^\infty (\Omega-E_k)\).

Examples and remark

Example 2.1 Given any set \(\Omega\), both \(\{\emptyset, \Omega\}\) and \(2^\Omega\) are \(\sigma\)-algebras on \(\Omega\). The set of all subsets \(E\) such that \(E\) is countable or \(\Omega-E\) countable is a \(\sigma\)-algebra (countable union of countable set is countable, else if one element’s complement is countable, then the union’s complement is countable, too).

Remark. A \(\sigma\)-algebra on \(\Omega\) is a subset of \(2^\Omega\) that contains \(\emptyset\) and is closed under countable union. Compare to a topology on \(\Omega\), which is a subset of \(2^\Omega\) that contains \(\emptyset, \Omega\), and is closed under arbitrary unions and finite intersections.

Definition 2.3 (measurable space, measurable set) A measurable space is a pair \((\Omega, \Sigma)\) where \(\Sigma\) is a \(\sigma\)-algebra on \(\Omega\). A subset \(S\subset \Omega\) is \(\Sigma\)-measurable iff \(S\in \Sigma\).

Definition 2.4 (freely generated σ-algebra) Given a set \(\Omega\) and \(\mathcal A\subset 2^\Omega\), the intersection of all \(\sigma\)-algebras \(\Sigma\) on \(\Omega\) such that \(\mathcal A\subset \Sigma\) is a \(\sigma\)-algebra on \(\Omega\).

In other words, every subset \(A\subset 2^\Omega\) freely generates a \(\sigma\)-algebra \(\sigma_{\mathrm{Alg}}(A)\) on \(\Omega\).

Definition 2.5 (monotone class) A monotone class \(\mathcal M\) is a set of sets with the following properties: \[\begin{align} A_1, \cdots \in \mathcal M, \quad A_j\nearrow A &\implies A\in \mathcal M \\ A_1, \cdots \in \mathcal M, \quad A_j\searrow A &\implies A\in \mathcal M \end{align}\]

Theorem 2.1 (monotone class theorem) Given a set \(\Omega\) and an algebra (definition 2.1) \(\mathcal A\) of subsets of \(\Omega\) such that \(\Omega, \emptyset\in \mathcal A\). The smallest monotone class \(\mathcal M\supset \mathcal A\) is \(\sigma_{\mathrm{Alg}}(\mathcal A)\).

Proof idea: every \(\sigma\)-algebra is trivially a monotone class. Monotone class and closure under finite union and complement implies \(\sigma\)-algebra. Show that \(\mathcal M\) is closed under finite union and complements.
Proof If a monotone class \(\mathcal M\supset \mathcal A\) is closed under complement and finite unions, then for any \(A_1, \cdots\in \mathcal A\), the sequence of partial unions approaches the countable union in limit: \[ \left(B_j = \bigcup_{k=1}^j A_j \in \mathcal M\right) \nearrow \bigcup_{j=1}^\infty A_j \in \mathcal M \] Together with complements, this shows that that \(\mathcal M\) is a \(\sigma\)-algebra. To show that \(\mathcal M\) is indeed closed under complement and finite union, argue for every set \(A\in \mathcal M\) that the set \(C(A)\) of all sets which stay within \(\mathcal M\) under union is a monotone class: \[ A\cup \bigcup_{j=1}^\infty B_j = \bigcup_{j=1}^\infty A\cup B \in \mathcal M \] By the algebraic property of \(\mathcal A\) we have \(\mathcal A\subset C(A) =\mathcal M\) (this step requires the minimality of \(\mathcal M\)!). This shows that \(\mathcal M\) is closed under complement and finite union, hence is a \(\Sigma\)-algebra.

Definition 2.6 (product σ-algebra) The product \(\sigma\)-algebra is the smallest \(\sigma\)-algebra containing all rectangles. \[ \Sigma = \sigma_{\mathrm{Alg}}(\Sigma_1\times \Sigma_2) = \sigma_{\mathrm{Alg}}\left(\{ A_1\times A_2 | A_1\in \Sigma_1, A_2\in \Sigma_2 \}\right) \] where here \(\Sigma_1\times \Sigma_2\) denotes the Cartesian product of sets. The product measure \(\mu = \mu_1\times \mu_2\) is

Corollary 2.1 Given an algebra of sets, its freely generated monotone class (smallest monotone class containing it) is also its freely generated \(\sigma\)-algebra.

Measure space

Definition 2.7 (measure, measure space) Given a measurable space \((\Omega, \Sigma)\), a measure on \((\Omega, \Sigma)\) is a function \(\mu:\Sigma\to [0, \infty]\) with \[ \mu \left(\bigsqcup_{k=1}^\infty E_k\right) = \sum_{k=1}^\infty \mu(E_k), \quad \mu(\emptyset) = 0 \] for every disjoint sequence \(\{E_j\} \in \Sigma\). A measure space is an ordered triple \((\Omega, \Sigma, \mu)\).

Proposition 2.2 (elementary properties of measures) Given a measure space \((\Omega, \Sigma, \mu)\) and \(D\subset E\in \Sigma\):

  1. \(\mu(D) \leq \mu(E)\).
  2. \(\mu(E - D) = \mu(E) - \mu(D)\).

Proposition 2.3 (measure of nested union) Under a measure space \((\Omega, \Sigma, \mu)\) with a nested chain \(E_1\subset E_2\subset \cdots\), we have \(\mu \bigcup_{j=1}^\infty E_j = \lim_{j\to \infty} \mu(E_j)\).

Proof Only consider the case \(\mu(E_k) < \infty\) and define \(E_0=\emptyset\), then \(\bigcup_{j=1}^\infty E_j = \bigsqcup_{j=1}^\infty E_j - E_{j-1}\). Using the disjoint union additivity \[\begin{align} \bigcup_{j=1}^\infty E_j &= \sum_{j=1}^\infty \mu(E_j - E_{j-1}) = \lim_{k\to \infty} \sum_{j=1}^k \mu(E_j) - \mu(E_{j-1}) = \lim_{k\to \infty} \mu(E_k) \end{align}\]

The dual of the proposition above is given below.

Corollary 2.2 (measure of nested intersection) Under a measure space \((\Omega, \Sigma, \mu)\) with a nested chain \(E_1\supset E_2\supset \cdots\), we have \(\mu \bigcap_{j=1}^\infty E_j = \lim_{j\to \infty} \mu(E_j)\).

Definition 2.8 (finite and σ-finite measures) A measure space \((\Omega, \Sigma, \mu)\) is \(\sigma\)-finite if there are countably many sets \(A_1, \cdots\) such that \[ \mu(A_{\forall j}) < \infty, \quad \Omega = \bigcup_j A_j \] with \(\sigma\)-finiteness, w.l.o.g. we can prove \(A_j\) to be disjoint. The measure space is finite if \(\mu(\Omega)<\infty\).

Theorem 2.2 (uniqueness of measures) Given \(\Omega\) and \(\Sigma = \sigma_{\mathrm{Alg}}(\mathcal A)\), where \(\mathcal A\) is a set algebra with \(\Omega, \emptyset \in \mathcal A\). If \(\mu_1\) is a \(\sigma\)-finite measure in the stronger sense such that \[ \exists \{A_j\in \mathcal A\}_{i=1}^\infty, \quad \mu_1(A_j)<\infty, \quad \bigcup_{j=1}^\infty A_j=\Omega \] Then every \(\mu_2\) which coincides with \(\mu_1\) on \(\mathcal A\) satisfies \(\mu_1=\mu_2\).

Proof First assume that \(\mu_1(\Omega)<\infty\). Consider \[ \mathcal M = \{A\in \Sigma: \mu_1(A) = \mu_2(A)\} \supset \mathcal A \] If this is shown to be a monotone class, then \(\Sigma\subseteq \mathcal M\) (in fact it’s equality by \(A\in \Sigma\) condition). Given an increasing sequence of sets \(A_j\subset \mathcal M\), construct \(B_1 = A_1, B_{j>1} = A_j - A_{j-1}\), then \[ \mu_1(A_{n\to \infty}) = \mu_1 \bigsqcup_{j=1}^{n\to \infty} B_j = \lim_{n\to \infty} \sum_{j=1}^n \mu_1(B_j) = \lim_{n\to \infty} \mu_1(A_n) = \lim_{n\to \infty} \mu_2(A_n) = \mu_2(A_{n\to \infty}) \] Additionally, \(A\in \mathcal A\implies A^c\in \mathcal M\) since \(\mu_j(A^c) = \mu_j(\Omega)-\mu_j(A)\) by \(\mu_j\) finite, from which \(\mathcal M\) is a monotone class. Next, for general \(\sigma\)-finite measures, \(\mu_1(B\cap A_0) = \mu_2(B\cap A_0)\) for every \(\mu(A_0)<\infty\) and \(B\in \Sigma\) since \(A_0\cap \Sigma = \sigma_{\mathrm{Alg}}(A_0 \cap \mathcal A)\). By \(\sigma\)-finite assumption choose \(\Omega = \bigsqcup A_j\) each with finite measure, then \[ \mu_1(B\in \Sigma) = \mu_1 \bigsqcup_{j=1}^\infty A_j\cap B = \sum_{j=1}^\infty \mu_1(A_j\cap B) = \sum_{j=1}^\infty \mu_2(A_j\cap B) = \mu_2(B) \]

Measurable function

Proposition 2.4 (composition preserves measurability) Given a measurable chain \[ (\Omega_1, \Sigma_1)\xrightarrow f (\Omega_2, \Sigma_2) \xrightarrow g (\Omega_3, \Sigma_3) \] then \(g\circ f\) is \(\Sigma_1\)-measurable.

We next prove a convenient criteria for proving measurability.

Proposition 2.5 (algebra of function preimages) The inverse image operation commutes with elementary algebraic operations: given \(f:\Omega\to Y\), then

  1. \(f^{-1}(Y-A) = \Omega - f^{-1}(A)\).
  2. \(f^{-1}(\bigcup_{A\in \mathcal A} A) = \bigcup_{A\in \mathcal A} f^{-1}(A)\).
  3. \(f^{-1}(\bigcap_{A\in \mathcal A} A) = \bigcap_{A\in \mathcal A} f^{-1}(A)\).

Proposition 2.6 (condition for measurable function) Given measurable spaces \((\Omega_1, \Sigma_1), (\Omega_2, \Sigma_2)\), such that \(\Sigma_2 = \sigma_{\mathrm{Alg}}(\mathcal A)\). For every function \(f:\Omega_1\to \Omega_2\) \[ f^{-1}(\forall A\in \mathcal A) \in \Sigma_1 \implies f \quad \Sigma_1\text{-measurable}. \]

Proof

Consider the subsets whose preimage are measurable \[ \mathcal T = \{B\in \Sigma_2 | f^{-1}(B)\in \Sigma_1\} \supset \mathcal A \] It suffices to show that \(\Sigma_2 \subset \mathcal T\). To do so, we show that \(\mathcal T\) is a \(\sigma\)-algebra by the well-behavior of set operations under preimage:

  • \(\emptyset = f^{-1}(\emptyset) = \emptyset \in \Sigma_1, f^{-1}(\Omega_2) = \Omega_1\in \Sigma_1\).
  • For complement: given \(B\in \mathcal T: f^{-1}(\Omega_2-B) = \Omega_1-f^{-1}(B) \in \Sigma_1\).
  • For countable union: \(\displaystyle f^{-1}\bigcup B_j = \bigcup f^{-1}B_j \in \Sigma_1\).
The algebra of preimage commutes with all elementary set operations.

Carathéodory theorem

Definition 2.9 (outer measure) Given a set \(\Omega\), a function \(\mu:2^\Omega\to \mathbb R_{\geq 0}\) is an outer measure if it satisfies:

  1. non-negativity: \(\mu(\emptyset) = 0\).
  2. monotonicity: \(A\subset B\implies \mu(A) \leq \mu(B)\).
  3. countable sub-additivity: for any countable collection of subsets \(A_j\subset \Omega\): \[ \mu \bigcup_{j=1}^\infty A_j \leq \sum_{j=1}^\infty \mu(A_j) \]

Definition 2.10 (Carathéodory criteria) Given an outer measure \(\mu:2^\Omega\to \mathbb R_{\geq 0}\), a set \(S\subset \Omega\) satisfies the Carathéodory criteria (for measurability) if \[ \mu(\forall E) = \mu(E\cap A) + \mu(E\cap A^c) \] The set \(S\) is also said to be measurable (by the Carathéodory criteria).

Lemma 2.1 (measurable sets form a set-algebra) The set \(\Sigma\subset 2^\Omega\) of measurable sets is an algebra of sets that contains \(\emptyset, \Omega\).

Proof We have \(\emptyset, \Omega \in \Sigma\) by definition 2.10 immediately. To show that \(\Sigma\) is closed under countable intersection: given \(A, B\in \Sigma\), applying the given condition on \((E, A), (\cdot, B)\) yields
\[\begin{align} \mu(E) &= \mu(E\cap A) + \mu(E\cap A^c) \\ &= \mu(E\cap A\cap B) + \mu(E\cap A\cap B^c) + \mu(E\cap A^c\cap B) + \mu(E\cap A^c \cap B^c) \end{align}\] We wish to show that the RHS equals \(\mu(E\cap A\cap B) + \mu(E\cap (A\cap B)^c)\). To this end, note that (using addition in place of union to reduce clutter): \[\begin{align} (A\cap B)^c &= A^c\cap B + A\cap B^c + A^c\cap B^c \\ \mu(E\cap (A\cap B)^c) &= \mu(E\cap (A\cap B)^c\cap A) + \mu(E\cap (A\cap B)^c \cap A^c) \\ &= \mu(E\cap A\cap B^c) + \mu(E\cap A^c\cap B + E\cap A^c\cap B^c) \\ &= \mu(E\cap A\cap B^c) + \mu(E\cap A^c\cap B) + \mu(E\cap A^c\cap B^c) \\ \end{align}\]

The following lemmas each depend on the previous one and culminates in the Carathéodory theorem.

Lemma 2.2 (countable additivity on measurable sets) Given disjoint sets \(B_1, \cdots \in \Sigma\): \[ \mu \left( E\cap \bigsqcup_{j=1}^\infty B_j \right) = \sum_{j=1}^\infty \mu(E\cap B_j) \] In particular, \(\mu\) is countable additive on \(\Sigma\) by setting \(E=\Omega\): \[ \mu \bigsqcup_{j=1}^\infty (B_j\in \Sigma) = \sum_{j=1}^\infty \mu(B_j) \]

Proof For disjoint \(B_1, B_2\in \Sigma\), applying definition 2.10 yields \(\mu(B_1\sqcup B_2) = \mu(B_1) + \mu(B_2)\); finite additivity follows by induction. In particular, this implies the following limit since it holds for each \(m\) \[ \lim_{m\to \infty} \mu\left( E\cap \bigsqcup_{j=1}^m B_j \right) = \sum_{j=1}^\infty \mu(E\cap B_j) \tag{2.1} \] We next show countable additivity: given disjoint \(B_1, \cdots\) \[ \lim_{m\to \infty} \mu\left( E\cap \bigsqcup_{j=1}^m B_j \right) \leq \mu \left( E\cap \bigsqcup_{j=1}^\infty B_j \right) = \mu \left(\bigsqcup_{j=1}^\infty E\cap B_j\right) \leq \sum_{j=1}^\infty \mu(E\cap B_j) \] But the LHS and RHS are exactly those of equation (2.1), so we obtain the equality by sandwich-ing \[ \mu \left( E\cap \bigsqcup_{j=1}^\infty B_j \right) = \sum_{j=1}^\infty \mu(E\cap B_j) \]

Lemma 2.3 (Carathéodory-measurable sets form a sigma-algebra) \(\Sigma\) is a \(\sigma\)-algebra; equivalently in light of lemma 2.1, \[ B_{\forall j}\in \Sigma\implies \bigcup_{j=1}^\infty B_j \in \Sigma \]

Proof Without loss of generality assume \(B_j\) distinct. One direction follows from countable subadditivity: \[ \mu(E) = \mu((E\cap B)\cup (E\cap B^c)) \leq \mu(E\cap B) + \mu(E\cap B^c) \] For the other direction, the following equation holds for each \(m\) thus in limit \[ \mu_E = \lim_{m\to \infty} \mu\left(E\cap \bigcup_{j=1}^m B_j \right) + \mu \left[ E\cap \left(\bigcup_{j=1}^m B_j\right)^c \right] \tag{2.2} \] The limit of the first term is known to be \(\mu(E\cap B)\), the second term \[ \mu \left[ E\cap \left(\bigcup_{j=1}^m B_j\right)^c \right] \geq \mu(E\cap B^c) \] by subadditivity; substituting into equation (2.2) completes the proof. \[ \mu_E \geq \mu\left(E\cap \bigcup_{j=1}^\infty B_j \right) + \mu \left[ E\cap \left(\bigcup_{j=1}^\infty B_j\right)^c \right] \]

Theorem 2.3 (Carathéodory theorem) Given an outer measure \(\mu:2^\Omega\to \mathbb R_{\geq 0}\):

  1. The collection \(\Sigma\subset 2^\Omega\) of subsets satisfying Carathéodory’s criterion is a \(\sigma\)-algebra,
  2. \(\mu:\Sigma\to \mathbb R_{\geq 0}\) is a countably additive measure on \(\Sigma\).