My class notes define a signed measure on a measurable space ( X , R ) as a σ-additive function ν : R → R . (I take this to mean we're only considering finite measures.) I'm confused on what I'm supposed to interpret σ-additive to mean here. My first guess would be that if A = ⋃ n A n where A 1 , A 2 , … are disjoint, then ν ( A ) = ∑ n = 1 ∞ ν ( A n ) ;i.e., just the usual definition of σ-additivity. But this seems problematic when the series on the right doesn't converge absolutely, because then its value depends on the order of the A i , which my gut tells me shouldn't be the case. Does σ-additivity here also involve the claim that the series on the right always converges absolutely? The wikipedia page on signed measures does require this, but no other source I found online, or my class notes, explicitly states it.

Question

My class notes define a signed measure on a measurable space   (  X  ,      R    ) as a   σ-additive function   ν  :      R    →      R  . (I take this to mean we&#039;re only considering finite measures.) I&#039;m confused on what I&#039;m supposed to interpret   σ-additive to mean here. My first guess would be that if   A  =      ⋃    n        A    n   where       A    1    ,      A    2    ,  … are disjoint, then  ν  (  A  )  =      ∑          n      =      1        ∞    ν  (      A    n    )  ;i.e., just the usual definition of   σ-additivity. But this seems problematic when the series on the right doesn&#039;t converge absolutely, because then its value depends on the order of the       A    i  , which my gut tells me shouldn&#039;t be the case. Does   σ-additivity here also involve the claim that the series on the right always converges absolutely? The wikipedia page on signed measures does require this, but no other source I found online, or my class notes, explicitly states it.

britspears523jp · Accepted Answer

ν  takes only finite values. So disjointness of   (      A    n    ) implies that   ∑  ν  (      A    n    ) is a convergent series. So is any rearrangement of terms since disjointness still holds. If a series of real numbers converges whenever the terms are permuted then the series is absolutely convergent. Hence   ∑      |    ν  (      A    n    )      |    &amp;lt;  ∞.

My class notes define a signed measure on a measurable space ( X , <mrow class="MJX-TeX

Answered question

Answer & Explanation

New Questions in Pre-Algebra