Extending pre-measures to measures

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Stub grade: A*
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Warning:This page is currently being written, the problem of extending a pre-measure on a ring of sets, [ilmath]\mathcal{R} [/ilmath] 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.


TODO: Fill this in

Proof steps

  1. A pre-measure, [ilmath]\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath], can be extended to an outer-measure, [ilmath]\mu^*:\mathcal{H}_{\sigma R}(\mathcal{R})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath]
  2. the set of all [ilmath]\mu^*[/ilmath]-measurable sets forms a ring
  3. the set of all [ilmath]\mu^*[/ilmath]-measurable sets forms a [ilmath]\sigma[/ilmath]-ring
  4. An outer-measure is countably additive on the [ilmath]\sigma[/ilmath]-ring of all [ilmath]\mu^*[/ilmath]-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.