Q1) Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, $X\colon \Omega \to \mathbb R$ a r.v. and $\mathcal G$ a subalgebra of $\mathcal F$. Could someone explain me the meaning of $$\mathbb P\{X\leq x\mid \mathcal G\}\ \ ?$$ i.e., the distribution of $X$ given $\mathcal G$.
Q2) Let $(\Omega \times \tilde \Omega , \mathcal F\times \mathcal A,\mathbb P)$ a probability space and $X:\Omega \times \Omega '\longrightarrow \mathbb R$ a r.v. Could someone explain me the meaning of $$\mathbb P\{X\leq x\mid \mathcal A\} \ \ ?$$
I don't really understand those type of conditional probabilities.