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*probability measure*on same space [itex]\Omega[/itex] is a function of subsets of [itex]\Omega[/itex] satisfying three axioms:

(i) For every set [itex]A \subset \Omega[/itex], the value of the function is a non-negative number: P(A) [itex]\geqslant[/itex] 0.

(ii) For any two disjoint sets A and B, the value of the function for their union A + B is equal to the sum of its value for A and its value for B:

P(A + B) = P(A) + P(B) provided A.B = [itex]{\O}[/itex].

(iii) The value of the function for [itex]\Omega[/itex] (as a subset) is equal to 1:

P([itex]\Omega[/itex]) = 1.

**Now, reply these questions:**

If

*M*is a probability measure, show:

(a) that the function

*M*/2 satisfies Axiom(i) and (ii) but not (iii).

(b) the function [itex]M^2[/itex] satisfies (i) and (iii) but not necessary (ii); give a counterexample to (ii).