Some Properties of Sigma Algebra

lemma: Let $X : \Omega \rightarrow \Omega'$ . Then $(X^{-1} a)^C = X^{-1} a^C$ when $a$ is a subset of $\Omega'$ and $X^{-1} \cup A = \cup X^{-1} A$ when $A$ is a function into subsets of $\Omega'$ .

$q \in (X^{-1} a)^C$ iff $q \not \in X^{-1} a$ iff $X(q)\not \in a$ iff $X(q) \in a^C$ iff $q \in X^{-1} a^C$ .

$q \in X^{-1} \cup A$ iff there is $\lambda$ such that $q \in X^{-1} A(\lambda)$ iff $X(q) \in A(\lambda)$ iff $q \in X^{-1} A(\lambda)$ iff $q \in \cup X^{-1} A$ .

lemma: Let $X : \Omega \rightarrow \Omega'$ and let $A$ be a system of subsets of $\Omega'$ . Then $\sigma X^{-1} A = X^{-1} \sigma A$

Let $Q$ be a system of subsets of $\Omega'$ such that $q \in Q$ iff $X^{-1} q \in \sigma X^{-1} A$ . Then $A \subseteq Q$ . Furthermore, $Q$ is a sigma algebra of $\Omega'$ :

$X^{-1} \Omega' = \Omega \in \sigma X^{-1} A$ .
if $q \in Q$ , then $X^{-1} q \in \sigma X^{-1} A$ . But then $X^{-1} q^C = (X^{-1} q)^C \in \sigma X^{-1} A$ and so $q^C \in Q$ .
if $\left< q_i \right>$ is a sequence of sets of $Q$ then $\left< X^{-1} q_i \right>$ is a sequence of sets of $\sigma X^{-1} A$ and so $X^{-1} (\cup_i q_i) = \cup_i X^{-1} q_i \in \sigma X^{-1} A$ and so $\cup_i q_i \in Q$ .

Hence $\sigma A \subseteq Q$ and so $X^{-1} \sigma A \subseteq \sigma X^{-1} A$ . Since $X^{-1} A \subseteq X^{-1} \sigma A$ and $X^{-1} \sigma A$ is a sigma algebra, it follows that $\sigma X^{-1} A \subseteq X^{-1} \sigma A$ . Hence $\sigma X^{-1} A = X^{-1} \sigma A$ .

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