Measure theoretic reason why i.i.d. random variables have the same expectation?

Question

Let $(\Omega, \mathfrak{F}, \mathbb{P})$ be a probability space and $X, Y$ two independent and identically distributed real random variables.

Does it hold that $\forall A \in \mathfrak{F}$ we have $$\mathbb{E}[X : A] = \mathbb{E}[Y : A] $$

Answer 1

I assume that by $\mathbb E[X:A]$ you mean the conditional probability usually denoted as $\mathbb E[X|A]$.

The statement is false. Take $X$ and $Y$ two i.i.d Bernoulli variables with $p\neq1$ and consider the event $A=\{X=1\}$. Then $\mathbb E[X|A]=1$, but by independence $\mathbb E[Y|A] = \mathbb E[Y] = p \neq 1$.