How would you prove the law of total probability for random variables?
The Theorem states: Let $X: \Omega \mapsto S$ and $Y:\Omega \mapsto S'$ be random variables. Then, for all $ A\subseteq S', P (Y ∈ A) =\sum \nolimits_{ x∈X(Ω):P(X=x)>0} P (Y ∈ A|X = x) P (X = x)$.
My initial idea goes: $A=(Y\in A) \cup ({X=x})$ such that $x$ partitions $S$. Then $P(A)=\sum\nolimits_{x\in X} P(Y \in A)P(X=x)$, as $S$ and $S'$ are disjoint, so $P(A)=\sum\nolimits_{x\in X:P(x)>0} P((Y \in A)|X=x))P(X=x)$