Can Lebesgue Analysis Exchange Sums of Finite Terms?

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SUMMARY

In Lebesgue analysis, it is established that finite sums can be interchanged with integrals without concerns for convergence. Specifically, integrals are linear, allowing for the manipulation of finite sums as if they were infinite sums with only a finite number of nonzero terms. This principle simplifies the analysis of finite sums within integrals, making it straightforward to take sums outside of integrals when dealing with finite terms.

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natski
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Hi all,

I'm aware that in Lebesgue analysis, one can interchange between the sum of infinite terms of an integral to the integral of a sum of infinite terms,... but what about a sum of finite terms? If the sum goes to N rather than infinity, can a sum inside an integral still be taken outside?

Natski
 
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That is even easier.
-integrals are linear
-no need to consider convergence
-you could write the finite sum as an infinite sum with only a finite number of nonzero terms
 

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