Dominated Convergence Theorem

Suppose is a measure space. Suppose is a sequence of measurable functions s.t. pointwise.

If a non-negative integrable function st

Then

Proof Since is integrable, are integrable.

Note:-

Apply Fatou’s Lemma to to get

Since is integrable, , we can subtract from the both sides of the above inequality to get

Now,

If is a sequence of integrable fns on st

Then,

and

Proof

Set . Then by MCT

This implies is real valued a.e. ie converges a.e

Thus converges almost everywhere.

Denote , . Then a.e. and

So since is integrable apply DCT to get

Proposition

Suppose , where

and is integrable for each

and let

a) If there exist an integrable fn st

and

then is continuous.

b) If exists and integrable fn s.t

Then is differentiable and

Example: Let be integrable, , then

define

Prove that is continuous.

Take be a sequence converging to .

Let be any point in . Then Denote

Then

Apply DCT,

Note: