关于测度在‘不可测’映射下的像成为测度的注记

A Note on Measures Transported by `Non-measurable' Mapping

设(Ω,F)与(E,ε)是两个可测空间,μ是(Ω,F)上的任一非零测度.以μ表μ的外测度,Aμ表Ω上的μ可测集全体,Fμ表F关于μ的完备化.设是从Ω到E的任一映射.若∶(Ω,Aμ)→(E,ε)是可测的,则μ(-1(·))是(E,ε)上的一个测度.反之,即使∶Ω→(E,ε)不是Aμ可测的,μ(-1(·))仍可以是(E,ε)上的一个测度;进一步μ(-1(·))是(E,ε)上的一个σ有限测度的充要条件是∶Ω→(E,ε)是Fμ可测(当然更是Aμ可测)的且μ是σ有限的.

Let(Ω,F) and(E,ε) be two measurable spaces, and μ a nonzero measure on(Ω,F). Let μ* be the outer measure of μ, A μ* the set of all μ*-measurable sets on Ω and Fμ the completion of F with respect to μ. Suppose  is an arbitrary map from Ω to E clearly, if :(Ω,A μ*)→(E,ε) is measurable, then μ*( -1(·)) is a measure on(E,ε). On the contrary, even if ∶Ω→(E,ε) is not A*μ-measurable,μ*( -1(·))can also be a measure on(E,ε);furthermore,μ*( -1(·))is a-finite measure on(E,ε) if and only if  is Fμ-measurable and μ is σ-finite.