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Afklare forholdet mellem ydre foranstaltninger
Dette er relateret til min målteori klasse, men det er ikke lektier. Motivationen bag dette indlæg er at forstå det nøjagtige forhold mellem de tre objekter er nævnt i overskriften.
Carathéodory s sætning hedder det, givet en ydre foranstaltning $ \ varphi $ på en plads $ X $, det sæt af $ \ varphi $-målelige sæt danner en $ \ sigma $-algebra og $ \ varphi $ begrænset til de målbare sættene er et tælleligt tilsætningsstof komplet foranstaltning. Jeg spekulerer på, om dette indrømmer et omvendt:
Hver foranstaltning $ \ mu $ induceres af en ydre foranstaltning $ \ mu ^ {\ ast} $ canonically opnået fra $ \ mu $, men generelt $ \ sigma $-algebra af $ \ mu ^ {\ ast} $ - målbare sæt kan være strengt større end den, hvor $ \ mu $ er defineret. Men de $ \ mu $-målelige sæt er de samme som de $ \ mu ^ {\ ast} $-målelige sæt, forudsat at $ \ mu $ er
fuldføre (naturligvis nødvendigt) og
$ \ Sigma $-finit (ikke strengt nødvendigt, men tilstrækkelig og god nok til mange formål)
og dermed $ \ mu $ kan fås tilbage fra $ \ mu ^ {\ ast} $ ved Carath konstruktion (eller blot begrænsning, hvis du foretrækker det).
De $ \ sigma $-finiteness tilstand kan blive svækket til localizability / lokale bestemmelse forhold, men jeg foretrækker at formulere resultatet for $ \ sigma $-finite foranstaltninger, da dette er et mere udbredt tilstand.
Jeg bogføring dette, fordi jeg synes det er helt relevant for mit spørgsmål, selv om det ikke direkte løse det. Selvfølgelig trivielt nul foranstaltning virker for alle målbare rum $ (X, \ Sigma) $. Lidt mere interessant,
For en vilkårlig $ x_0 \ i X $, at foranstaltningen \ mu_ {x_0} (A) = \ begin {sager} 1, \ i A, \ \ 0, \ notin A, \ end {sager} er en sandsynlighed foranstaltning ( derfor også $ \ sigma $-finite).
Foranstaltningen $ $
\ Mu (a) = \ begin {sager}
\ Infty,converse taske, \ neq \ Emptyset,hvide converse,
\ \ 0, = \ Emptyset,
\ End {sager} foreslået af Niels er en komplet foranstaltning, men den undlader at være $ \ sigma $-finite (og undlader helt dårligt også). Selvfølgelig, teknisk dette besvarer mit andet spørgsmål,køb converse online.

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