View source for Extending pre-measures to measures

{{Stub page|grade=A*|msg={{Warning|This page is currently being written, the problem of extending a pre-measure on a [[ring of sets]], {{M|\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==
{{Todo|Fill this in}}
==Proof steps==
# A [[extending pre-measures to outer-measures|pre-measure, {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } }}, can be extended to an outer-measure, {{M|\mu^*:\mathcal{H}_{\sigma R}(\mathcal{R})\rightarrow\overline{\mathbb{R}_{\ge 0} } }}]]
# [[The set of all mu*-measurable sets forms a ring|the set of all {{M|\mu^*}}-measurable sets forms a ring]]
# [[The set of all mu*-measurable sets forms a sigma-ring|the set of all {{M|\mu^*}}-measurable sets forms a {{sigma|ring}}]]
# [[An outer-measure is countably additive on the sigma-ring of all mu*-measurable sets|An outer-measure is countably additive on the {{sigma|ring}} of all {{M|\mu^*}}-measurable sets]]
# [[Every set of outer-measure 0 belongs to the set of all mu*-measurable sets]]
# [[The outer-measure is a complete measure on the set of all mu*-measurable sets]] (called the [[measure induced by an outer-measure]])
# [[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.
==References==
<references/>
{{Measure theory navbox|plain}}
{{Theorem Of|Measure Theory}}