Prove why this equality holds

45 Views Asked by Bumbble Comm At 02 Apr 2026 - 5:38

If the sum is finite there is no problem , but if it is not how i can prove or show that the following happens

$\sum_{n=1}^{\infty}n^p\mu(E_n)$ =$\int\sum_{n=1}^{\infty}n^p\chi_{E_n}$

Also , if someone can help me ending the proof of this problem , it would be apreciated so much

There are 2 best solutions below

Bumbble Comm On 14 Nov 2014 - 2:22

Where are you stuck? Remember that an infinite sum is defined to be the limit of a finite sum...

Bumbble Comm On 14 Nov 2014 - 2:27

Apply the monotone convergence theorem to the partial sums of the sum in the integrand.