Let X and Y be discrete r.v.s and recall (from the application following Theorem 11) that B'Y

Question:

Let X and Y be discrete r.v.s and recall (from the application following Theorem 11) that B'Y is the (-field of subsets of ( defined by B'Y = {B' ( (; Y-1(B') = A for some A ( A}. For x, y ( ( with P(Y = y) > 0, consider the conditional probability
P(X = x|Y = y) = P(X = x, Y = y) / P(Y = y)
And show that, for each fixed x, the function P(X = x | Y = .) is B'Y-measurable?