## B. Extension of an Outer Measure to a Probability Measure

To use the outer measure as a probability measure for more general sets that those in F0, we have to extend the algebra F0 to a a-algebra F of events for which the outer measure is a probability measure. In this appendix it will be shown how Fcan be constructed via the following lemmas.

Lemma 1.B.1: For any sequence Bn of disjoint sets in ^, P*(U=j Bn) <

£Г=1 p "(Bn).

Proof: Given an arbitrary є > 0 it follows from (1.21) that there exists a countable sequence of sets An, j in F0 such that Bn c An, j and P *(Bn) >

ZjU P(An, j) – є2 n; hence,

CO CO CO CO CO CO

£ P*(Bn) > £ £ P(An, j) – £ £ 2-n = £ £ P(An, j) – є. n=1 n=1 j=1 n=1 n=1 j=1

(1.28)

Moreover, UO=1 Bn c UO=1 UO=1 An, j, where the latter is a countable union of sets in Ж0; hence, it follows from (1.21) that

( |

CO CO

и BA...

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