Promise of formal definition of conditional expectation: what is $E[X|Y=y]$ exactly?

  1. $E[X | \sigma(Y)$ is a RV measurable w.r.t. $\sigma(Y)$, which you seem to be OK with.

  2. Any random variable $Z$ which is measurable w.r.t. $\sigma(Y)$ is essentially a function of $Y$, i.e., there exists a measurable $f$ such that $Z = f(Y)$ almost surely.

Putting 1 and 2 together, there exists a measurable function $g$ such that $E[X | \sigma(Y)] = g(Y)$ almost surely. You can think of $E[X | Y=y]$ as a shorthand for $g(y)$.

($g$ is of course not unique and determined up its a.s. equivalence class)

Tags:

Probability

Real Analysis

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