What does different conditional mean

Question

Could someone explain the difference in the meaning between these:

$$E(X|Y)$$

$$E(X|Y=y)$$

$$E(X|y)$$

Answer 1

Here are some:

• The mathematical object E[X|Y] is a random variable, it exists as soon as X and Y are defined on the same probability space and X is integrable.
• The mathematical object E[X|Y=y] is a number, it exists as soon as X and Y are defined on the same probability space and [Y=y] has positive probability.
• The mathematical object E[X|y] does not exist.

Answer 2

One way to see it is the following: We know by a "fundamental lemma" of probability (or rather measure theory) that any $\sigma(Y)$-measurable function can be represented as a function of $Y$. This is the case for the conditional expectation and thus
$$ E(X | Y) = f(Y) $$ With this in mind we can define the conditional expectation in a point as $$ E(X | Y = y) \stackrel{\text{def}}{=} f(y) $$

Then we have that if $\mu$ is the law of $Y$ we have that this newly defined function satisfies $$ E(X) = E( E(X | Y ) ) = E(f(Y)) = \int_{\mathbb{R}} f(y) \, \mu(dy) = \int_{\mathbb{R}} E(X | Y = y) \, \mu(dy) $$