1
$\begingroup$

Let $ (\Omega, \mathcal A, \operatorname{P})$ be a probability space, with a real random variable $X$ and a sub-σ-algebra $ \mathcal B \subseteq \mathcal A$

In probability theory with measure theoretical treatment, conditional expectation $E(X| \mathcal{B})$ is defined first, and then conditional probability $P(A| \mathcal{B}), A \in \mathcal{A}$ is defined as $E(I_A| \mathcal{B})$.

There is another commonly used conditional probability given an event $P(A|C):=\frac{P(A \cap C)}{P(C)}, A, C \in \mathcal{A}, P(C) >0$. Can it be defined from $E(X| \mathcal{B})$ or $P(A| \mathcal{B})$?

For example, let $C \in \mathcal{A}, P(C) > 0$, and $\sigma(C):=\{\emptyset, C, \bar{C}, \Omega\}$.

is $P( | \sigma(C) )$ regular? That is, is $P( | \sigma(C) )(\omega)$ a probability measure?

is $P( | \sigma(C) )(\omega)$ same for all $\omega \in C$, and same for all $\omega \in \bar{C}$?

is $\frac{P(A \cap C)}{P(C)} = P( A | \sigma(C) )(\omega)$ for all $\omega \in C$ and $A \in \mathcal A$?

Thanks!

$\endgroup$
4
$\begingroup$

Whenever $\mathsf P(C)>0$ you have $$ \mathsf P(A|\sigma(C))(\omega) = 1_C(\omega) \mathsf P(A|C) + 1_{\bar C}(\omega)\mathsf P(A|\bar C). $$ which you can easily check by the definition of the conditional expectation. Thus, for any $\omega$ you have $\mathsf P(\cdot|\sigma(C))(\omega)$ is either of two probability measures $\mathsf P(\cdot|C)$ or $\mathsf P(\cdot|\bar C)$ which positively answers your question.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.