Law of total probability given it belongs to A

Question

Given that $P_y(y) =\sum P_{y|x}(y|x)·P_x(x)$ Law of total probability. Prove that for any given event A $P_y(y∈A) =∑P_{y|x}(y∈A|x)·P_x(x)$

I tried to define y∈A and tried solving it using the compliment of the event. I am not able to arrive at the intended outcome

Answer 1

I tried to define $y\in A$ and tried solving it using the compliment of the event.

You do not need the compliment.   You need the additivity of disjoint unions.   For any discrete random variable, $Y$, and set of measures, $A~$:

$$\mathsf P_{\small Y}(A)~=~\sum_{y\in A}\mathsf P_{\small Y}(y)$$