Do signed measures on delta-rings always have a Hahn decomposition? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:40:23Z http://mathoverflow.net/feeds/question/89998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89998/do-signed-measures-on-delta-rings-always-have-a-hahn-decomposition Do signed measures on delta-rings always have a Hahn decomposition? Ricky Demer 2012-03-01T23:33:18Z 2012-03-02T00:43:36Z <p>(aka, what I'd <em>meant</em> to ask <a href="http://mathoverflow.net/questions/89839/do-signed-measures-on-sigma-rings-always-have-a-hahn-decomposition" rel="nofollow">here</a>) <br><br><br><br><br> Let $X$ be a set. $\:$ Let $\mathcal{R}$ be a non-empty subset of $2^X$. $\:$ Suppose $\mathcal{R}$ is a <a href="http://en.wikipedia.org/wiki/Delta-ring" rel="nofollow">delta-ring</a>. <br><br><br> Let $\: \phi : \mathcal{R} \to \mathbb{R} \:$ be such that for all sequences $\: \langle A_0,A_1,A_2,A_3,...\rangle \:$ of pairwise disjoint elements of $\mathcal{R}$, <br><br> if $\;\;\;\; \displaystyle\bigcup_{n=0}^{\infty} \; A_n \;\; \in \;\; \mathcal{R} \;\;\;\;$ then $\;\;\;\; \displaystyle\sum_{n=0}^{\infty} \; \phi(A_n) \;\; = \;\; \phi\left(\displaystyle\bigcup_{n=0}^{\infty} \; A_n\right) \;\;\;\;$. <br><br></p> <blockquote> <p>Does it follow that there exists a subset $P\hspace{.02 in}$ of $X$ (which is not necessarily a member of $\mathcal{R}$) <br> such that for all members $A$ of $\mathcal{R}$, $\;\;\; A\cap P \: \in \: \mathcal{R} \;\;$ and $\; \phi(A-P\hspace{.02 in}) \leq 0 \leq \phi(A\cap P\hspace{.02 in}) \;\;\;$?</p> </blockquote> <p><br></p>