# Approximating Borel Sets

1. Dec 11, 2013

### jeterfan

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

We're given the measure space $$(X,A,μ)$$ with $$X=\bigcup_{i=1}^{\infty} X_i$$ where $$X_i⊂X_{i+1}⊂..., X_i$$ are open for all i and μ(X_i)<+∞ for all i. Show that for every Borel set B there exists an open set U where μ(B\U)<ϵ.

2. Relevant equations

Pick closed sets $$c_i⊂X_i−B$$ with $$μ((X_i−c_i)−B)=μ((X_i−B)−c_i)<ϵ/{2^i}$$. Let $$G=\bigcup_i^\infty X_i-c_i$$. Then recognizing that $$X_i−c_i=X_i⋂c_i^c$$ is the intersection of open sets reveals that G must be open. Moreover, B⊂G is clear and thus we can consider G−B. If follows that μ(G−B)<ϵ by sub additivity.
My problem is explaining why such closed sets exist. I know that $$X_i-B$$ is measurable, but why am I able to find a closed set inside of it that satisfies these requirements?