Can the Beppo-Levi relation explain moving sums out of integrals?

[stukbv]

My lecturer has said that beppo levi means for and increasing sequence of X_i where X_i is simple for all i, it holds that

∫lim_{i → ∞}X_idP = lim_{i → ∞}∫X_idP

But why is it that he later says things like

∫ lim_{i→ ∞} Ʃⁱ_n=1P₂(B_w1n)dP₁(w1) = lim_{i → ∞}Ʃⁱ_n=1∫P₂(B_w1n)dP₁(w1)
is a result of beppo levi? Where in beppo levi does it say you can move the sum out of the integral?!

 

[oli4]

Hi stukbv
The serie (sum) is also a sequence, if the terms you are summing over are positives (or 0) then the sum of the terms is an increasing sequence
therefore the theorem implies that you can take the sum outside of the integral for the same reason that you could take the limit outside before.
That is, your sequence of increasing terms is defined by Ui=Ʃ(up to i)(positive terms)

Cheers...

 

[stukbv]

I see, thata makes sense. Thanks for your help