A Treatise on Measurability

Throughout this writeup, let $\Omega, \Psi$ be sets and $X : \Omega \to \Psi$ be a function. Refer to a previous post for prior measure-theoretic notions that will be used in this set of exercises.

Problem 1. For any $\sigma$ -algebra $\mathcal G$ on $\Psi$ , define $\mathcal G_{X^{-1}}$ by

$\mathcal G_{X^{-1}} := \{X^{-1}(M) : M \in \mathcal G\}.$

Prove that $\mathcal G_{X^{-1}}$ is a $\sigma$ -algebra on $\Omega$ , known as the pull-back $\sigma$ -algebra of $X$ .

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Solution. We have $\emptyset = X^{-1}(\emptyset) \in \mathcal G_{X^{-1}}$ and $\Omega = X^{-1}(\Psi) \in \mathcal G_{X^{-1}}$ .

The complementation and union properties follows from set algebra via

$\displaystyle \begin{aligned} \Omega \backslash X^{-1}(M) &= X^{-1} (\Omega \backslash M) \in \mathcal G_{X^{-1}}, \\ \bigcup_{i=1}^\infty X^{-1}(V_i) &= X^{-1} \left( \bigcup_{i=1}^\infty V_i \right) \in \mathcal G_{X^{-1}}. \end{aligned}$

Problem 2. For any $\sigma$ -algebra $\mathcal F$ on $\Omega$ , define $\mathcal F_X$ by

$\mathcal F_X := \{M \in \mathcal P(\Psi) : X^{-1}(M) \in \mathcal F\}.$

Prove that $\mathcal F_X$ is a $\sigma$ -algebra on $\Psi$ , known as the push-forward $\sigma$ -algebra of $X$ .

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Solution. Follow the proof in Problem 1.

Let $\mathcal F$ be a $\sigma$ -algebra on $\Omega$ and $\mathcal G$ be a $\sigma$ -algebra on $\Psi$ .

Definition 1. A function $W : \Omega \to \Psi$ is said to be $\mathcal F$ / $\mathcal G$ –measurable if for any $M \in \mathcal G$ , $W^{-1}(V) \in \mathcal F$ .

Problem 3. Prove that $X$ is $\mathcal F$ / $\mathcal G$ -measurable if and only if $\mathcal G_{X^{-1}} \subseteq \mathcal F$ . Since this holds for any $\sigma$ -algebra $\mathcal F$ , we say that $\mathcal G_{X^{-1}}$ is the coarsest $\sigma$ -algebra such that $X$ is measurable.

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Solution. We first prove $(\Rightarrow)$ . Suppose $X$ is $\mathcal F$ / $\mathcal G$ -measurable. Fix $L \in \mathcal G_{X^{-1}}$ . Then $L = X^{-1}(M)$ for some $M \in \mathcal G$ , which implies $L = X^{-1}(M) \in \mathcal F$ , as required.

Next, we prove $(\Leftarrow)$ . Suppose $\mathcal G_{X^{-1}} \subseteq \mathcal F$ . Fix $M \in \mathcal G$ . By definition, $X^{-1}(G) \in \mathcal G_{X^{-1}} \subseteq \mathcal F$ . Therefore, $X$ is $\mathcal F$ / $\mathcal G$ -measurable.

Problem 4. Prove that $X$ is $\mathcal F$ / $\mathcal G$ -measurable if and only if $\mathcal G \subseteq \mathcal F_X$ . Since this holds for any $\sigma$ -algebra $\mathcal G$ , we say that $\mathcal F_X$ is the finest $\sigma$ -algebra such that $f$ is measurable.

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Solution. We first prove $(\Rightarrow)$ . Suppose $X$ is $\mathcal F$ / $\mathcal G$ -measurable. Fix $M \in \mathcal G$ . By measurability, $X^{-1}(M) \in \mathcal F$ , so that $M \in \mathcal F_X$ .

Next, we prove $(\Leftarrow)$ . Suppose $\mathcal G \subseteq \mathcal F_X$ . Fix $M \in \mathcal G$ . By assumption, $M \in \mathcal G \subseteq \mathcal F_X$ , so that $X^{-1}(M) \in \mathcal F$ . Therefore, $X$ is $\mathcal F$ / $\mathcal G$ -measurable.

Problem 5. For any $\Lambda \subseteq \Omega$ , prove that $\mathcal F_{\iota^{-1}} = \{\Lambda \cap L : L \in \mathcal F\}$ , where $\iota$ denotes the inclusion map.

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Solution. Each element in $\mathcal F_{\iota^{-1}}$ is of the form $\iota^{-1}(L)$ for some $L \in \mathcal F$ . Since $\iota^{-1}(L) \subseteq \Lambda$ and $\iota^{-1}(L) \subseteq L$ , we have $\iota^{-1}(L) \subseteq \Lambda \cap L$ . The result follows by noticing $\iota(\Lambda \cap L) \subseteq L \Rightarrow \Lambda \cap L \subseteq \iota^{-1}(L)$ .

Problem 6. Let $(\Phi, \mathcal H)$ be a measurable space. Suppose $Y : \Psi \to \Phi$ is $\mathcal G$ / $\mathcal H$ -measurable. Prove that $Y\circ X : \Omega \to \Phi$ is $\mathcal F$ / $\mathcal H$ -measurable.

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Solution. Fix $N \in \mathcal H$ . Since $Y$ is $\mathcal G$ / $\mathcal H$ -measurable, $Y^{-1}(W) \in \mathcal G$ . Since $X$ is $\mathcal F$ / $\mathcal G$ -measurable, $(Y \circ X)^{-1}(N) = X^{-1}(Y^{-1}(N)) \in \mathcal F$ . Therefore, $g \circ f$ îs $\mathcal F$ / $\mathcal H$ -measurable.