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

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1 March 15
- 1.1 A measurable set?

[edit] March 15

[edit] A measurable set?

Let $(X,μ)$ and $(Y,ν)$ be measure spaces and ${A k},{B k}$ sequences of sets of finite measure in X and Y respectively. Let the "rectangles" $R_k = A_k \times B_k$ , and assume that

$\sum_{k=1}^\infty (\mu \otimes \nu)(R_k) < \infty.$

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

$T_n = \{x\ |\ 1/n \leq \sum_k \nu(R_{k,x})\}.$

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

Well,

R k, x

is just B_k if x is in A_k, and empty otherwise. So

∑	ν(R_k,x)
k

is the some of the measures of the B_k such that x is in A_k. Thus whether x is in T_n is determined by which of the A_ks x is in, and T_n is a union of intersections of the A_ks. Algebraist 17:45, 15 March 2008 (UTC)

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

$V_k = \{x\ |\ f_1(x) + \cdots + f_k(x) > \alpha\}$