Problem 3: Let Q be a sample space, and P a probability on 9. Suppose that 31,32,33 are pairwise disjoint, have positive probability, and their union is Q. We let g I: 0' (Bl,Bz,B3) . For a xed set A C Q, suppose that we have ]P(A|g) = llBT Here, H32 denotes the indicator function of the set .32. (a). Show that the random variable Y := P (A | g) is g-measurable. Let .7: := 20 (the power set of (2), and let H := {0, 9}. (b). Is Y .F-measurable? Why? (c). Is Y 'H-measurable? Why