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  1. In elementary probability,

    $E(Y \mid X =x)$ is defined as expectation of $Y$ w.r.t. the p.m. $P(A \mid X =x): = \frac{P(A \cap \{X=x \})}{P( X=x)}$ when $P( X=x) \neq 0$.

    when $P(X =x) =0$ , define $P(A\mid X =x): = \lim_{\epsilon \rightarrow 0} \frac{P(A \cap \{|X-x|< \epsilon\} )}{P(|X-x|< \epsilon)}$ and $E(Y \mid X =x): = \lim_{\epsilon \rightarrow 0} \frac{E(Y \times 1_{\{|X-x|< \epsilon\}})}{P(|X-x|< \epsilon)}.$

    If define $f(x):=P(A \mid X =x)$ and $h(x):=E(Y \mid X =x)$, then $f(X)$ and $h(X)$ are both random variables $\Omega \rightarrow \mathbb{R}$.

  2. In probability theory, $E(Y \mid X )$ and $P(A \mid X )$ are both random variables $\Omega \rightarrow \mathbb{R}$.

I was wondering

  1. if $h(X)$ and $E(Y \mid X )$ are the same a.s.?

  2. Similarly, if $f(X)$ and $P(A \mid X )$ are the same a.s.? Is $f(X=x)$ the so called transition probability distribution?

Thanks and regards!

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    @Didier: (4) I know$P(A|X=x)$and$E(Y|X=x)$do not exist in general, but I consider cases when they exist. However ill-posed my questions are, my intention is to find connection between, when exist, conditional probabilities/expectations defined in non-measure theoretical courses/books and those in measure-theoretical courses/books.2011-05-07

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Answer to both questions: Yes if the limit exists.

ps. don't forget to replace t by small x or x by t in your formulas.

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    are there some references that talk about this connection?2011-05-07