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Sigma-Algebra, Measures, Properties of Measures

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Let m be a sigma-algebra, M_1 and M_2 are measures on m.

a). Is M = M_1 + M_2 a measure?

b). Is M = M_1 - M_2 a measure?

c). Is M = M_1M_2 a measure?

Either prove or disprove by providing a counter example.

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Solution Summary

Sigma-Algebra, Measures, Properties of Measures are investigated.

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a) True
For any X in m, we know M(X)=M_1(X)+M_2(X). If X is an empty set, then M_1(X)=M_2(X)=0 and thus M(X)=0. Since M_1 and M_2 are measures and they satisfy the countable additivity, thus M also satisfy the countable ...

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