# Complete measure space

1. Jan 22, 2008

### johnson123

[SOLVED] complete measure space

1. The problem statement, all variables and given/known data

Assume that ($$\Omega$$,$$\Sigma$$,$$\mu$$) is a complete
measure space, let $$\mu_{e}$$ be the outer measure defined by $$\mu$$
. Prove that if $$\mu_{e}$$(S)=0 $$\Rightarrow$$ S$$\in$$$$\Sigma$$ .

I know that $$\mu_{e}$$ = $$\mu$$ when restricted on $$\Sigma$$

and that if $$\mu$$(A)=0, then every subset of A has measure zero from completeness.

I intuitively can see this to be true, since if S where not in $$\Sigma$$ and
$$\mu_{e}$$ = 0 , then this would contradict the fact that ($$\Omega$$,$$\Sigma$$,$$\mu$$) is complete, since the fact that
$$\mu_{e}(A_{1})$$=$$\mu_{e}(A_{2})$$...=$$\mu_{e}(A_{n})$$=0, for $$A_{1}\subseteq A_{2}\subseteq.......\subseteq A_{n}\subseteq S$$
would imply that $$\Sigma$$ is incomplete, but not sure how to rigorously prove
it.

Last edited: Jan 22, 2008