Wikipedia / Outer measure
Wikipedia / Outer measure
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.
that fulfills the following requirements:
.
have the same measure
Countable additivity: for any sequence of pairwise disjoint subsets
It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
Outer measures
denote the collection of all subsets including the empty set An outer measure is a set function such that
If, instead, an outer measure were allowed to take negative values, its definition would have to be modified to take into account the possibility of non-convergent infinite sums.
An alternative and equivalent definition. such that
Measurability of sets relative to an outer measure
is -measurable (sometimes called Carathéodory-measurable, after the mathematician Carathéodory
-measurable subset is one which may be used as a building block, breaking any other subset apart into pieces (namely, the piece which is inside of the measurable set together with the piece which is outside of the measurable set). In terms of the motivation for measure theory, one would expect that area are disjoint subsets of the plane. However, the formal logical development of the theory shows that the situation is more complicated. A formal implication of the axiom of choice is that for any definition of area as an outer measure which includes as a special case the standard formula for the area of a rectangle, there must be subsets of the plane which fail to be measurable. In particular, the above "expected principle" is false, provided that one accepts the axiom of choice.
The measure space associated to an outer measure
-measurability to see that
-measurable then its complement-measurable.
The following condition is known as the "countable additivity on measurable subsets."
A similar proof shows that:
-measurable.
The properties given here can be summarized by the following terminology: This measure space has the additional property of completeness, which is contained in the following statement:
-measurable.
This is easy to prove by using the second property in the "alternative definition" of outer measure.
Restriction and pushforward of an outer measure
.
Pushforward
by
.
Restriction
Let be a subset of . Define by
One can check directly from the definitions that is another outer measure on .
Measurability of sets relative to a pushforward or restriction
If a subset of is -measurable, then it is also -measurable for any subset of .
Given a map and a subset of , if is -measurable then is -measurable. More generally, is -measurable if and only if is -measurable for every subset of .
Regular outer measures
Definition of a regular outer measure
Given a set , an outer measure on is said to be regular can be approximated 'from the outside' by -measurable sets. Formally, this is requiring either of the following equivalent conditions:
There exists a -measurable subset of which contains .
The regular outer measure associated to an outer measure
Given an outer measure on a set , define by
Then is a regular outer measure on which assigns the same measure as to all -measurable subsets of . Every -measurable subset is also -measurable, and every -measurable subset of finite -measure is also -measurable.
So the measure space associated to may have a larger σ-algebra than the measure space associated to . The restrictions of and to the smaller σ-algebra are identical. The elements of the larger σ-algebra which are not contained in the smaller σ-algebra have infinite -measure and finite -measure.
From this perspective, may be regarded as an extension of .
Outer measure and topology
Suppose is a metric space and an outer measure on . If has the property that
whenever
then is called a metric outer measure.
Theorem. If is a metric outer measure on , then every Borel subset of is -measurable. (The Borel setss of are the elements of the smallest -algebra generated by the open sets.)
Construction of outer measures
There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as Method I and Method II.
Method I
Let be a set, a family of subsets of which contains the empty set and a non-negative extended real valued function on which vanishes on the empty set.
Theorem. Suppose the family and the function are as above and define
That is, the infimum extends over all sequences of elements of which cover , with the convention that the infimum is infinite if no such sequence exists. Then is an outer measure on .
Method II
The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose is a metric space. As above is a family of subsets of which contains the empty set and a non-negative extended real valued function on which vanishes on the empty set. For each , let
and
Obviously, when since the infimum is taken over a smaller class as decreases. Thus
exists (possibly infinite).
Theorem. is a metric outer measure on .
This is the construction used in the definition of Hausdorff measures for a metric space.