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Question about probability measure

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An 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).
 

Answers and Replies

Hurkyl
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We can't help if you if you haven't tried it.
 
sic
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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)
 

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