Lecture 21 - Mar 7

is a premeasure on an algebra on over . Let be the Carathedory extension of on .

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Theorem

If with , then given there exists such that .

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NOTE Take a measure on . Define

Then is a metric on . Exercise:-Prove

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Proof

Fix , we first get such that and

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NOTE Definition for outer measure

,

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As , there exists such that

Take

Now,

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Remark:- is equivalent to

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NOTE Can product of measure be a measure?

Example , ,

In general,

and

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Definition (Regular)

Let be a topological space and is a measure. The measure is called regular if for every and there exists an open set and closed set such that and and is an -algebra containing .

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Proposition

If is regular then

Proof

Let . By regularity, for every , there exists open set and closed set such that

Define

Then

Hence,

We can write,

As ,

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Theorem

Suppose is a metric space and is a measure, then

1. For every ,

2. If then same holds for .

3. If is -compact i.e. there exists countably many compact sets such that Then