I was given the following question. I know how to answer the first part, but I am confused on the last part... "Show that, for each.." How do I answer this last part?

Problem 19.3. Let X = Y = [0, 1], M = Bo,1, N = 2, let / Lebesgue measure on M, and let v counting measure on . Let A denote the diagonal A = {(x, x) : x E [0, 1] } C [0, 1] x [0, 1]. Prove that A E M O N and 1E(x, y) du(x) | du(y) (57) X are unequal. Show that, for each P E M O N, the functions f(x) = v(Px) and g(y) = M(Py) are measurable (with respect to M and N respectively) and T(P) = | v(Px) du, p(P) = u(Py) du X Y are both measures M ON. (Note that Theorem 15.7 does not (directly) apply)