Difference between revisions of "Mu*-measurable set"

Revision as of 20:44, 24 May 2016

Definition

Given an outer-measure, [ilmath]\mu^*:H\rightarrow\bar{\mathbb{R} }_{\ge 0} [/ilmath] (for a hereditary sigma-ring, [ilmath]H[/ilmath]) we define a set, [ilmath]A\in H[/ilmath] as [ilmath]\mu^*[/ilmath]-measurable if^[1]:

• [ilmath]\forall B\in H\big[\mu^*(B)=\mu^*(B\cap A)+\mu^*(B-A)\big][/ilmath]^{[Note 1]}

See also

• The set of all [ilmath]\mu^*[/ilmath]-measurable sets is a ring - an important step on the way to restricting an outer-measure to a measure

TODO: More links, also link to page for the restriction of outer measure to measure directly, once such a page exists

Notes

1. ↑ Halmos gives a great abuse of notation here, by writing [ilmath]B\cap A'[/ilmath] (where [ilmath]A'[/ilmath] denotes the complement of [ilmath]A[/ilmath]), of course in a ring of sets (sigma or not) we do not have a complementation operation, only set subtraction

References

1. ↑ Measure Theory - Paul R. Halmos