Lecture 06 - Jan 16

We have a - additive ser function (pre - measure) on , Then we defined

is an algebra.

Lemma 4

If is disjoint then where .

Proof

For , there is nothing to prove, assume holds for

Take

is a disjoint collection.

Lemma 5

is a -algebra.

Proof

It is enough to prove that is closed under countable disjoint unions because

If then we define

Since is an algebra

Take a disjoint collection.

Write

Take , then

Taking we get

NOTE

Then

Lemma 6

Proof

follows from the definition of .

So we want to prove . It is enough to prove

such that

Take any a cover of . Set

Then is disjoint and

Then,

Lemma 7

is a measure.

Proof

For any disjoint collection , then

NOTE

By Lemma 4,

Taking , then we have reverse inequality.

, a -additive set function (pre-measure) on an algebra. Then a -algebra containing and measure

NOTE

NOTE

• Borel -algebra

• Lebesgue -algebra

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