Extending pre-measures to measures

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This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Warning:This page is currently being written, the problem of extending a pre-measure on a ring of sets, $\mathcal{R}$ to a measure is not trivial. For example, to find the biggest class of sets we can extend a pre-measure to is different to what this page shows. This page is just starting to be put together.

Statement

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TODO: Fill this in

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Proof steps

1. A pre-measure, $\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} }$, can be extended to an outer-measure, $\mu^*:\mathcal{H}_{\sigma R}(\mathcal{R})\rightarrow\overline{\mathbb{R}_{\ge 0} }$
2. the set of all $\mu^*$-measurable sets forms a ring
3. the set of all $\mu^*$-measurable sets forms a $\sigma$-ring
4. An outer-measure is countably additive on the $\sigma$-ring of all $\mu^*$-measurable sets
5. Every set of outer-measure 0 belongs to the set of all mu*-measurable sets
6. The outer-measure is a complete measure on the set of all mu*-measurable sets (called the measure induced by an outer-measure)
7. Every set in the sigma-ring generated by a ring of sets is mu*-measurable

Is a good path I think. I need to develop this page more after I've cleaned up some of the existing notes pages.