When exactly does $E(X|Y)$ mean? (X, Y are random variable)

Question

Am I correct that $E(X|Y)$ is "expected value of $X$ conditional on each possible value of $Y$"?

I have this question because I wondered that $E(X|Y)$ means "expected value of $X$ conditional on all possible value of $Y$ altogether". However, then it sounds impossible to me that $E(X)$ can be unequal to $E(X|Y)$ even when $X$ and $Y$ are dependent of each other.

Answer 1

Let $X,Y$ be random variables defined on the same probability space.

Then $E[X\mid Y]$ is the notation of a random variable with the following properties:

• $E[X\mid Y]$ is measurable wrt $\sigma(Y)$
• For every $A\in\sigma(Y)$ we have: $$\int_A X(\omega)P(d\omega)=\int_A E[X\mid Y](\omega)P(d\omega)$$

Often - if we are aiming to find $E[X\mid Y]$ - we can do that by calculating $f(y):=\mathbb E[X\mid Y=y]$ and then draw the conclusion that $E[X\mid Y]=f(Y)$.

In the special case where $X$ and $Y$ are independent this results in a degenerated random variable: $\mathbb E[X\mid Y](\omega)=\mathbb EX$ for every $\omega\in\Omega$.