# Wikipedia:Reference desk/Archives/Mathematics/2008 March 15

Mathematics desk
< March 14 << Feb | March | Apr >> March 16 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

# March 15

## A measurable set?

Let ${\displaystyle (X,\mu )}$ and ${\displaystyle (Y,\nu )}$ be measure spaces and ${\displaystyle \{A_{k}\},\{B_{k}\}}$ sequences of sets of finite measure in X and Y respectively. Let the "rectangles" ${\displaystyle R_{k}=A_{k}\times B_{k}}$, and assume that

${\displaystyle \sum _{k=1}^{\infty }(\mu \otimes \nu )(R_{k})<\infty .}$

Let ${\displaystyle R_{k,x}=\{y\ |\ (x,y)\in R_{k}\}}$ and

${\displaystyle T_{n}=\{x\ |\ 1/n\leq \sum _{k}\nu (R_{k,x})\}.}$

Why is it obvious that Tn is measurable?  — merge 17:01, 15 March 2008 (UTC)

Well, ${\displaystyle R_{k,x}}$ is just Bk if x is in Ak, and empty otherwise. So ${\displaystyle \sum _{k}\nu (R_{k,x})}$ is the some of the measures of the Bk such that x is in Ak. Thus whether x is in Tn is determined by which of the Aks x is in, and Tn is a union of intersections of the Aks. Algebraist 17:45, 15 March 2008 (UTC)

Oh, I think I see how it works out. If ${\displaystyle \{f_{k}\}}$ is a sequence of nonnegative measurable real-valued functions and α is a real number, the sets

${\displaystyle V_{k}=\{x\ |\ f_{1}(x)+\cdots +f_{k}(x)>\alpha \}}$

are measurable, and so are

${\displaystyle W_{\alpha }=\bigcup _{k}V_{k}=\{x\ |\ \sum _{k=1}^{\infty }f_{k}(x)>\alpha \}}$

and

${\displaystyle \bigcap _{j}W_{\alpha -1/j}=\{x\ |\ \sum _{k=1}^{\infty }f_{k}(x)\geq \alpha \}}$.

In this case ${\displaystyle f_{k}(x)=\nu (R_{k,x})=\chi _{A_{k}}(x)\nu (B_{k})}$ and ${\displaystyle \alpha =1/n}$.  — merge 22:30, 16 March 2008 (UTC)