Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{G} \subset \mathcal{F}$ a sub-sigma algebra. Let $X$ and $Y$ be two random variables (with respect to $\mathcal{F}$).
What does the statement: "The distribution of $X$ given $\mathcal{G}$ is equal in law to $Y$" mean formally?
Does it mean that for all non-negative measurable functions $f : \mathbb{R} \rightarrow \mathbb{R}_+$ we have (with $E$ denoting $P$-expectations)
$$E(f(X)| \mathcal{G}) = E(f(Y)| \mathcal{G}) \quad ? $$