Conditional expectation is defined as follows:

We are given probability space $(\Omega, \Sigma, P)$

For $a \in \Sigma$ such that $P(A)>0$, random variable $X: \Omega \to \mathbb{R}$ we define: $$\mathbb{E} (X | A) = \int_{\Omega}X(\omega) \mathrm{d}P(\omega|A).$$

But there is also another definition: for $P(A)>0$, $$\mathbb{E} (X | A) = \frac{1}{P(A)} \int_{A}X(\omega) \mathrm{d}P.$$

However, I have trouble proving that those definitions are equivalent.

Could you tell me why the two formulas give the same result?


If $P(A)>0$ then $P(B|A)=\frac{P(A \cap B)}{P(A)}$, as you saw in elementary probability. So if $X=\chi_C$ then

$$\int_\Omega X(\omega) dP(\omega|A) = \frac{P(C \cap A)}{P(A)} = \frac{1}{P(A)} \int_A X dP.$$

Now extend to simple functions and finally random variables (as usual).


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