Conditional Expectations $E[h(X,Y)|X]$ and $E[h(x,Y)|X]$ agree at $X = x$?

Question

I am having trouble understanding $E[\cdot |X = x]$. Now I know measure theory and the theory of conditional expectations decently well. So I understand that $E[h(X,Y) | X]$ is a random variable of the form $g(X)$ and it makes sense to call $g(x)$ $E[h(X,Y) | X = x]$.

Now I have a couple of questions regarding this. First, the function $g(x)$ is only determined on the range of $X$. So does this mean that anytime anyone writes $E[\cdot | X= x]$ they are implicitly requiring that $x$ is indeed a value achieved by $x$? Next, it is sometimes be the case that $[X=x]$ is a null set and since $E[\cdot | X]$ is only determined upto null sets itself, does that cause any issues?

Also, and this is my main question, people often freely replace $E[h(X,Y) | X=x]$ with $E[h(x,Y)|X=x]$. Can you give me a proof of why this is valid?

Answer 1

$E( \cdot |X=x)$ is just a notation.For the last question, when $X$ and $Y$ are independent,$E[h(X,Y)|X]=(E[h(x,Y)])_{x=X}=g(X)$.So $E[h(X,Y)|X=x]=E[h(x,Y)]=g(x)$

Answer 2

It may be a little better to think of $E(h(X, Y) \mid X = x)$ as the value of $E(h(X, Y) \mid X)$ on the set $\{X = x\}$. If $P(X = x) = 0$, then, since conditional expectations are only defined up to a.s. equivalence, the value assigned to $E(h(X, Y) \mid X = x)$ could be any real number, and can be modified at will. You would just be dealing with different a.s. equivalent versions of $E(h(X, Y) \mid X)$.

To prove that $E(h(X, Y) \mid X = x) = E(h(x, Y) \mid X = x)$, use $$E(h(X, Y) \mid X)1_{\{X = x\}} = E(h(X, Y)1_{\{X = x\}} \mid X) = E(h(x, Y)1_{\{X = x\}} \mid X) = E(h(x, Y) \mid X)1_{\{X = x\}}$$ which means that $E(h(X, Y) \mid X) = E(h(x, Y) \mid X)$ a.s. on the set $\{X = x\}$.