For my bachelor thesis, I've been studying iterated random functions and a very limited amount of measure theory to understand it rigorously. One thing I could not understand is the following:
Suppose $S$ is a complete, seperable metric space and equip this with its Borel sigma-algebra. Let $P$ be a probability measure on this measurable space. Now given a Borel subset $B \subset S$ and $\epsilon > 0$, then there exists a compact subset $C \subset B$ such that $P(B) < P(C) + \epsilon$.
I don't understand why such $C$ must exist. To be honest, I haven't got a clue how to even look for such $C$, or why $B$ would even contain a non-trivial compact subset. Thanks in advance for any help!