Suppose is a non-negative integrable function on and .

Show that .

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- Oct 23rd 2009, 10:18 PMproblemLebesgue integral
Suppose is a non-negative integrable function on and .

Show that . - Oct 24th 2009, 12:24 AMMoo
Hello,

Let

We have (*)

Let's study

Since is a decreasing sequence, and hence is a non-negative integrable and increasing sequence.

So we can apply Lebesgue monotone convergence theorem, and we have .

Hence by making the limit as n tends to infinity in (*), we have

And hence the result...

Another direct way would've been to use Lebesgue's dominated convergence theorem... :

Note that , which is integrable.

And since ,

And hence the result.