Lecture 22 - Mar 10

Theorem

Let be a metric space and let

be a measure.

Then:

1. For every ,

and

2. If , then the same approximation formulas hold for .

3. If is -compact, then for every ,

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Proof (Part 1: Borel sets)

Define

We show that is a -algebra.

Step 1:

This is immediate from definitions.

Step 2: Closed under complements

Assume . Fix .

From approximation for , choose:

• open with ,
• closed with .

Hence

Now and

So

This gives outer and inner approximation for , hence .

Step 3: Closed under countable unions

Let and put

Fix .

For each , choose open such that

Let

Then is open, , and

Similarly, for each , choose closed such that

Define

Then

Since finite unions of closed sets are closed, define

Using continuity of measure and , for large we have

So has closed inner approximation and open outer approximation, hence .

Therefore is a -algebra.

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Step 4: Every closed set belongs to

Let be closed. For , define

Then each is open, , and .

Claim:

Proof of claim: if , then for each there exists with

Hence . Since is closed, .

Now by continuity from above,

Thus for any , for some ,

So closed sets satisfy the outer approximation formula, and therefore closed sets are in .

Since is a -algebra containing all closed sets, it contains

But by definition , hence

So Part 1 is proved.

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Remarks for Parts 2 and 3

1. For , write with and for some Borel null set . Using monotonicity and , the same inf-sup approximation identities transfer from to .

2. If is -compact, write

with compact. Combining inner approximation by closed sets with truncation on yields approximation from below by compact subsets, giving

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Completion details for Parts 2 and 3 were left as exercises in class.