An probability measure on same space $\Omega$ is a function of subsets of $\Omega$ satisfying three axioms:

(i) For every set $A \subset \Omega$, the value of the function is a non-negative number: P(A) $\geqslant$ 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 = ${\O}$.

(iii) The value of the function for $\Omega$ (as a subset) is equal to 1:

P($\Omega$) = 1.

If M is a probability measure, show:

(a) that the function M/2 satisfies Axiom(i) and (ii) but not (iii).

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

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Hurkyl
Staff Emeritus
Gold Member
We can't help if you if you haven't tried it.

David
sic
Omega={A,B}
M(gurnisht)=0
M({A})=1/2
M({B})=1/2
M({A,B})=1
then
M^2({A,B})!=M^2({A})+M^2({B})
(Either it's that simple or i don't understand the problem)