Q3. Let (X, B and (Y, C) be measurable spaces and (X x Y, B x C) be the product measurable space. Let 11 be a probability measure on (X x Y, B x C). Define the marginals of 11 to be the measures VI on X and v2 on Y defined by VI (B) = "(B x Y) and v2(c) = "(X x C), for B B and C e C respectively. is called a joining of VI and v2. (a) Show that VI x and deduce that the Radon-Nikodfin derivative is defined. (b) If f (x, y) is measurable on X x Y, write a formula for f x x Y fdll in terms of f x dvl and dv2 (c) Under which conditions is VI x v2? Give an example where this fails.