Suppose i have . How to show that the converges absoutely almost everywhere ,

f= , and .

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- Dec 9th 2011, 09:46 PMyounhockA.e Convergent Problem.
Suppose i have . How to show that the converges absoutely almost everywhere ,

f= , and . - Dec 9th 2011, 10:40 PMJose27Re: A.e Convergent Problem.
All of this is a consequence of Minkowski's integral inequality, we have ( is the counting measure on ):

Since the left hand side is finite, so is the right side, but an integrable function can only be infinite on a null set so we get that is finite a.e. The rest follows directly from the inequality above as well.