Measures, outer measures and measurable sets I

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Although I have taught Measure and Integration many times over the years, typically I don't have time to prove all of the details of the Carathéodory theory of outer measures and their associated $\sigma$-algebras and measures.

I've been thinking about how we prove some of the fiddly bits at the end, and trying to decide how to make some of the final bootstrapping from finite unions to infinite unions a bit more natural.

One possibility I have thought of is to change the order in which we prove some of the results, essentially in order to avoid proving some facts twice. However, this might be at the cost of having to prove some results again for general measures that we first prove for measures coming from outer measures.

On the other hand, every measure on a $\sigma$-algebra is actually the restriction to that $\sigma$-algebra of  a measure that comes from some outer measure! The same is also true for the more general "measures" on rings of sets and on semi-rings of sets (which need to be defined carefully). So in the next few posts I am going to explore different possible orders to prove the results we need to try to make the process easier to remember (and hopefully more natural).

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