Suppose is a non-negative integrable function on and .
Show that .
We have (*)
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.