My lecture notes define conditional expectation and independence as follows:
- Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G}$ be a sub-sigma field of $\mathcal{F}$. If $X$ is an integrable random variable, then the conditional expectation of $X$ given $\mathcal{G}$ is any integrable random variable $Z$ which satisfies the following two properties:
(CE1) $Z$ is $\mathcal{G}$-measurable.
(CE2) $$\forall \Lambda \in \mathcal{G}: \int_{\Lambda} Z \, d \mathbb{P}=\int_{\Lambda} X \, d \mathbb{P}$$
We denote $Z$ by $\mathbb{E}[X | \mathcal{G}]$.
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A finite family $\mathcal{G}_{1}, \ldots, \mathcal{G}_{n}$ of sub-sigma fields is independent if and only if $$\forall i \in \{1,\ldots,n\}: \Gamma_{i} \in \mathcal{G}_{i} \implies \mathbb{P}\left[\bigcap_{i =1}^n \Lambda_{i}\right]=\prod_{i =1}^n \mathbb{P}\left[\Lambda_{i}\right]$$
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For sub-sigma fields $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$, we denote by $\mathcal{G}_{1} \vee \mathcal{G}_{2}$ the smallest $\sigma$-field that contains $\mathcal{G}_{1} \cup\mathcal{G}_{2}$, i.e., $\mathcal{G}_{1} \vee \mathcal{G}_{2} = \sigma (\mathcal{G}_{1} \cup\mathcal{G}_{2})$.
My question:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X$ an integrable random variable. Let $\mathcal D,\mathcal G$ be sub-sigma fields of $\mathcal F$. Assume $\mathcal D$ is independent of $\sigma(X) \vee \mathcal G$.
For $A \in \sigma(X)$, I would like to ask if $\mathbb E [\mathbb{1}_A \mid \mathcal{D} \vee \mathcal{G}]$ is $\mathcal{G}$-measurable.
Please leave me just hints so that I can have a chance to practice. Thank you so much!
Best Answer
Yes, this is true. In fact $\mathbb E [\mathbb{1}_A \mid \mathcal{D} \vee \mathcal{G}]=\mathbb E [\mathbb{1}_A \mid \mathcal{G}]$. To prove this consider the class of all sets of the form $D \cap G$ where $D \in \mathcal{D}$ and $G \in \mathcal{G}$. This is a $\pi-$ system and it generates $\mathcal{D} \vee \mathcal{G}$. Since the collection of all sets $E$ such that $\int_E {1}_A dP =\int_E (E[\mathbb{1}_A \mid \mathcal{G})]dP$ is $\lambda-$ system we only have to verify that this equation holds when $E=D \cap G$. But this is an easy consequence of the independence assumption.