Remark:- If & are two non-negative measurable functions & then .

Theorem (MCT)

Suppose is a non-decreasing sequence of non-negative measurable functions s.t. . Then

Proof:- Since is also an increasing sequence of non-negative numbers, so its limit exists.

Moreover, . Therefore, .

To prove the reverse inequality, we need to show that

for all simple non-negative functions .

Take a such & . Then it is enough to prove:

Consider, .

Claim:- . Suppose . If then (because ).

If , then there exist such that

Then

Note:- is a increasing sequence, since and .

Define the measure on by

Then

Taking limit as in , we obtain

Take limit .

Proposition

Suppose are two measurable functions and non-negative functions. Then

use (Exercise)

Take sequence of simple non-negative measurable functions & s.t.

Then

But

By MCT,

If is a sequence of non-negative measurable functions and

Then,

Proof:- Let be the partial sum,

Then,

and as

But

Hence,

Let be a sequence of non-negative measurable fns. Then

Proof:- Define,

Then

But

Note:-