Talk:Singular measure

mutually singular

so why not call this article "mutually singular measures" like the first line says? --itaj 04:01, 12 May 2006 (UTC)

a question

let $(\Omega ,\Sigma ),$ be a measurable space. i'm talking here about real-value signed finite measures on this space. let M be a set of measures.

let N be the set of all measures absolutely continuous with respect to a measure in the linear span of measures in M. i.e. $N:=\{\nu :\exists \mu \in span(M)\ (\nu <<\mu )\}$

for a measure $\mu$ i'll denote ${singl}(\mu ):=\{\nu :\nu \perp \mu \}$. the set of measures mutually singular to $\mu$.

and for a set of measures L. ${Singl}(L):=\{\nu :\forall \mu \in L\ (\mu \perp \nu )\}$. the set of all measures mutually singular to all measures in L.

my question is if the above implies that $N=Singl(Singl(N))$ i know this is true if there's only one measure in M, but i need to know about infinite set M, countable and bigger. --itaj 04:05, 12 May 2006 (UTC)

If span is in the sense of vector spaces, then no. Consider $M:=\{\delta _{n}:n\in N\}$. Then $\sum 2^{-n}\delta _{n}\not \in span(M)$ or N, but it is in singl(singl(N)). (Cj67 23:39, 25 June 2006 (UTC))