The Unapologetic Mathematician

Mathematics for the interested outsider

Outer Measures

We’re going to want a modification of the notion of a measure. But before we introduce it, we have (of course) a few definitions.

First of all, a collection \mathcal{E}\subseteq P(X) of sets is called “hereditary” if it includes all the subsets of each of its sets. That is, if E\in\mathcal{E} and F\subseteq E, then F\in\mathcal{E} as well. It’s not very useful to combine this with the definition of an algebra, because an algebra must contain X itself; the only hereditary algebra is P(X) itself. Instead, we define a “ring” of sets (or a \sigma-ring) to be closed under union (countable unions for \sigma-rings) and difference operations, but without the requirement that it contain X; complements and intersections are also not guaranteed, since we built these from differences using X itself. Pretty much everything we’ve done so far with algebras can be done with rings, and hereditary \sigma-rings will be interesting objects of study.

Just like we found for algebras and monotone classes, the intersection of two hereditary collection is again hereditary. We can thus construct the “smallest” hereditary \sigma-ring containing a given collection \mathcal{E}, and we’ll call it \mathcal{H}(\mathcal{E}). In fact, it’s not hard to see that this is the collection of all sets which can be covered by a countable union of sets in \mathcal{E}; any \sigma-ring containing \mathcal{E} must contain all such countable unions, and a hereditary collection must then contain all the subsets.

Now, an extended real-valued set function \mu^* on a collection \mathcal{E} is called “subadditive” whenever E, F, and their union E\cup F are in \mathcal{E}, we have the inequality

\displaystyle\mu^*(E\cup F)\leq\mu^*(E)+\mu^*(F)

It’s called “finitely subadditive” if for every finite collection \{E_1,\dots,E_n\}\subseteq\mathcal{E} whose union is also contained in \mathcal{E} we have the inequality

\displaystyle\mu^*\left(\bigcup\limits_{i=1}^nE_i\right)\leq\sum\limits_{i=1}^n\mu^*(E_i)

and “countably subadditive” if for every sequence \{E_i\}_{i=1}^\infty of sets in \mathcal{E} whose union is also in \mathcal{E}, we have

\displaystyle\mu^*\left(\bigcup\limits_{i=1}^\infty E_i\right)\leq\sum\limits_{i=1}^\infty\mu^*(E_i)

Note that these differ from additivity conditions in two ways: we only ask for an inequality to hold, and we don’t require the unions to be disjoint.

Finally, we can define an “outer measure” to be an extended real-valued, non-negative, monotone, and countably subadditive set function \mu^*, defined on a hereditary \sigma-ring \mathcal{H}, and such that \mu^*(\emptyset)=0. Just as for a measure, we say that \mu^* is “finite” or “\sigma-finite” if every set has finite or \sigma-finite outer measure.

March 25, 2010 Posted by | Analysis, Measure Theory | 12 Comments

   

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