## Extending a Measure to an Outer Measure

Let be a measure in a ring (not necessarily an algebra) , and let be the hereditary -ring generated by . For every , define

That is, can be covered by a countable collection of sets in . For every such cover, sum up the -measures of all the sets in the cover, and define to be the greatest lower bound of such sums. Then is an outer measure, which extends to all of . Further, if is -finite, then will be too. We call the outer measure “induced by” .

First off, if itself, then we can cover it with itself and an infinite sequence of empty sets. That is, . Thus we must have . On the other hand, if is contained in the union of a sequence , then monotonicity tells us that , and thus . That is, must be equal to for sets ; as a set function, indeed extends . In particular, we find that .

If and are sets in with and is a sequence covering , then it must cover as well. Thus can be at most , and may be even smaller. This establishes that is monotonic.

We must show that is countably subadditive. Let and be sets so that is contained in the union of the . Let be an arbitrarily small positive number, and for each choose some sequence that covers such that

This is possible because the definition of tells us that we can find a covering sequence whose measure-sum exceeds by an arbitrarily small amount. Then the collection of all the constitute a countable collection of sets in which together cover . Thus we conclude that

Since was arbitrary, we conclude that

and so is countably subadditive.

Finally, if , we can pick a cover . If is -finite, we can cover each of *these* sets by a sequence so that . The collection of all the is then a countable cover of by sets of finite measure; the extension is thus -finite as well.

[...] and Induced Measures If we start with a measure on a ring , we can extend it to an outer measure on the hereditary -ring . And then we can restrict this outer measure to get [...]

Pingback by Measures and Induced Measures « The Unapologetic Mathematician | March 31, 2010 |

[...] -ring containing , and let be the smallest hereditary -ring containing . Given a measure on , it induces an outer measure on , which restricts to a measure on [...]

Pingback by Measurable Covers « The Unapologetic Mathematician | April 1, 2010 |

[...] existence is straightforward. We can induce an outer measure, and then restrict it to get . It’s straightforward to verify from the [...]

Pingback by Extensions of Measures « The Unapologetic Mathematician | April 6, 2010 |

[...] analogy with the outer measure induced on the hereditary -ring by the measure on the -ring , we now define the “inner [...]

Pingback by Inner Measures « The Unapologetic Mathematician | April 8, 2010 |

[...] a measure space , we will routinely use without comment the associated outer measure and inner measure on the hereditary -ring [...]

Pingback by Measurable Spaces, Measure Spaces, and Measurable Functions « The Unapologetic Mathematician | April 26, 2010 |