## Integral of Probability Distribution

I guess this question can go in either this section or analysis, it came form my stats class, so I'll put it here.

Let A = {a,b,c}, F = $2^A$ and P be a probability measure that that:
P({a})=1/2 P({b})=1/6 P({c})=2/6, then (A,F,P) is a probability space.

Define Y:A-->R as Y(w) = 1 if w = a or c and Y(w) = 0 if otherwise.

Let $P_Y$ be the probability distribution of Y

Find $\int y{P_Y(dy)}$

My problem is I'm not really sure what this integral represents. It looks like it should just be the expected value of Y, but I'd prefer to calculate it directly.

What I have is that since X(w) can only take 2 values, 0 and 1, we have that answer is $0*P_Y(0) + 1*P_Y(1) = 5/6$

Any thoughts?