Beppo-Levi relation to Sums

1. May 18, 2012

stukbv

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

∫limi → ∞XidP = limi → ∞∫XidP

But why is it that he later says things like

∫ limi→ ∞ Ʃin=1P2(Bw1n)dP1(w1) = limi → ∞Ʃin=1∫P2(Bw1n)dP1(w1)
is a result of beppo levi? Where in beppo levi does it say you can move the sum out of the integral?!

2. May 18, 2012

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...

3. May 18, 2012

stukbv

I see, thata makes sense. Thanks for your help