Difference between revisions of "Mu*-measurable set"

Latest revision as of 22:05, 20 August 2016

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-{{DISPLAYTITLE:{{M|\mu^*}}-measurable set}}
+#REDIRECT [[Outer splicing set]]
-{{Stub page|grade=A}}
-==Definition==
-Given an [[outer-measure]], {{M|\mu^*:H\rightarrow\bar{\mathbb{R} }_{\ge 0} }} (for a [[hereditary sigma-ring]], {{M|H}}) we define a set, {{M|A\in H}} as ''{{M|\mu^*}}-measurable'' if{{rMTH}}:
-* {{M|1=\forall B\in H\big[\mu^*(B)=\mu^*(B\cap A)+\mu^*(B-A)\big]}}<ref group="Note">Halmos gives a great abuse of notation here, by writing {{M|B\cap A'}} (where {{M|A'}} denotes the [[complement]] of {{M|A}}), of course in a [[ring of sets]] (sigma or not) we do not have a complementation operation, only set subtraction</ref>
-==See also==
-* [[The set of all mu*-measurable sets is a ring|The set of all {{M|\mu^*}}-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==
-<references group="Note"/>
-==References==
-<references/>
-{{Measure theory navbox|plain}}
 {{Definition|Measure Theory}}