Let $(\Omega, \mathcal{A},\mathbb{P})$ be a probability space and $Y:(\Omega, \mathcal{A})\rightarrow (\Omega', \mathcal{A}')$ be a random variable.
So usually conditional probability is defined via conditional expectation as $\mathbb{P}(A|Y):=\mathbb{E}[\chi_A|Y]$, where $\chi_A$ is the indicator function of the set $A$.
I am wondering now how the conditional distribution $\mathbb{P}(X|Y)$ for a random variable $X$ is defined via a conditional expectation? Since $X$ is not fixed to a specific set $A$ I dont know how to go on about this.