Hello,

Here is a solution (not from me)

Let's fix

Let

Because we need n in it...

We know that

.

We have

, except over

(this deals with inclusion of sets), which is a set which measure goes to 0 as n goes to infinity.

By taking the limit, we can say that

, except over a set of measure 0.

Thus it is true almost everywhere.

Since this is true for any

, we can take

and consider that

But the measure of this intersection is the measure of

Hence

almost everywhere.

And finally, we can use Lebesgue's dominated convergence theorem.