Hello,

Here is a proof..

Let be an increasing positive sequence that converges to , and where is a simple function.

We can then say that

Note that (this is very simple to prove with the above formula of )

Also note that saying is equivalent to saying since we work on positive functions.

__________________________________________

(because )

Therefore,

But is an increasing sequence. Hence is an increasing set.

Since (because remember that ), we have

Therefore

By the monotone convergence theorem, we can conclude :