Under what condition we can interchange order of a limit and a summation?

Suppose f(m,n) is a double sequence in R. Under what condition do we have lim? Thanks!

Answer

A fairly general set of conditions, sufficient for many applications, is given by the hypotheses of dominated convergence. (Note that sums are just integrals with respect to the counting measure on \mathbb{N}, so dominated convergence applies with no modification.)

Without domination, the idea is that lumps of positive mass can “escape to infinity” when one attempts to interchange sum and limit. Here is a basic example: let f_{m,n} = 1 if m = n and 0 otherwise. Then \sum_{m=1}^{\infty} f(m, n) = 1 for all n, so the LHS is 1, but \lim_{n \to \infty} f(m, n) = 0, so the RHS is 0. The point of domination is to prevent these lumps of mass from escaping.

Attribution
Source : Link , Question Author : zzzhhh , Answer Author : Qiaochu Yuan

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