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| ``Re: summation over arbitrary index set?''
by azdbacks4234 on 2008-09-02 10:49:38 |
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| I may write one in the next couple days. I would define it as follows (and I'm basically just ripping off Rudin): if A is any set and f:A-[0,infty] is a function into the non-negative extended real numbers, then the "sum"
\sum_{a\in A}f(a)
is defined to be
\sup\{sum_{a\in F}f(a):F\subseteq A,F finite\}. If you do it this way, then it turns out that the above "sum" is equal to the integral of f over A with respect to the counting measure. Then you can (naturally?) generalize to complex-valued functions in l^1(A) by defining the unordered sum to be the Lebesgue integral of the function over A. I suppose this may be a bit contrived, but I kind of like it. |
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