Extending premeasures to measures
From Maths
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This page is a stub, so it contains little or minimal information and is on a todo list for being expanded.The message provided is:
Warning:This page is currently being written, the problem of extending a premeasure 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 premeasure to is different to what this page shows. This page is just starting to be put together.
Statement
TODO: Fill this in
Proof steps
 A premeasure, [ilmath]\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath], can be extended to an outermeasure, [ilmath]\mu^*:\mathcal{H}_{\sigma R}(\mathcal{R})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath]
 the set of all [ilmath]\mu^*[/ilmath]measurable sets forms a ring
 the set of all [ilmath]\mu^*[/ilmath]measurable sets forms a [ilmath]\sigma[/ilmath]ring
 An outermeasure is countably additive on the [ilmath]\sigma[/ilmath]ring of all [ilmath]\mu^*[/ilmath]measurable sets
 Every set of outermeasure 0 belongs to the set of all mu*measurable sets
 The outermeasure is a complete measure on the set of all mu*measurable sets (called the measure induced by an outermeasure)
 Every set in the sigmaring 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.
References
