## Approximating Sets of Finite Measure

So, we’ve got a -finite measure on a ring , and we extend it to a measure on the -ring . But often it’s a lot more convenient to work with itself than the whole of . So, to what extent can we do this efficiently?

As it turns out, if has finite measure and , then we can find a set so that .

Any set can be covered by a sequence of sets in , and we know that

That is, we can find such a cover satisfying

But since is continuous, we see that

The sequence of numbers increases until it’s within of its limit. That is, there is some so that if we define to be the union of the first sets in the sequence, we have

But now we can find

And thus .

[...] ring generated by the is itself countable, and so we may assume that is itself a ring. But then we know that for every and for every positive we can find some ring element so that . Thus is a [...]

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