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 .