I guess this question can go in either this section or analysis... I'll put it here.

Let A = {a,b,c}, F = and P be a probability measure such 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 be the probability distribution of Y

Find

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

Any thoughts?