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

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March 15

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A measurable set?

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Let   and   be measure spaces and   sequences of sets of finite measure in X and Y respectively. Let the "rectangles"  , and assume that

 

Let   and

 

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

Well,   is just Bk if x is in Ak, and empty otherwise. So   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)[reply]

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

 

are measurable, and so are

 

and

 .

In this case   and  .  — merge 22:30, 16 March 2008 (UTC)[reply]