Consider a probability space $(\Omega, \mathcal B, \mathbb P)$, and 2 random variables $X,Y:\Omega\to\mathbb R$
Conditional expectation $E(X|Y)=E(X|\sigma(Y))$ can be defined as a function that is
$\sigma(Y)$ -measurable. I.e. there cannot be a subset $A$ of $\Omega$ on which $Y$ maps to a unique value, but where the conditional expectation does not.
$E(E(X|Y)\cdot 1_A)=E(X\cdot 1_A)$ For all $A\in \sigma(Y)$.
How do we prove that this defines a function that is unique almost-everywhere?