Outer measure

Template:Short description Template:More footnotes 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.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> 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.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in <math>\mathbb{R}</math> or balls in <math>\mathbb{R}^{3}</math>. One might expect to define a generalized measuring function <math>\varphi</math> on <math>\mathbb{R}</math> that fulfills the following requirements:

Any interval of reals <math>[a,b]</math> has measure <math>b-a</math>
The measuring function <math>\varphi</math> is a non-negative extended real-valued function defined for all subsets of <math>\mathbb{R}</math>.
Translation invariance: For any set <math>A</math> and any real <math>x</math>, the sets <math>A</math> and <math>A+x=\{ a+x: a\in A\}</math> have the same measure
Countable additivity: for any sequence <math>(A_j)</math> of pairwise disjoint subsets of <math>\mathbb{R}</math>

<math> \varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i).</math>

It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of <math>X</math> is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.

Outer measuresEdit

Given a set <math>X,</math> let <math>2^X</math> denote the collection of all subsets of <math>X,</math> including the empty set <math>\varnothing.</math> An outer measure on <math>X</math> is a set function <math display=block>\mu: 2^X \to [0, \infty]</math> such that

Template:Em: <math>\mu(\varnothing) = 0</math>
Template:Em: for arbitrary subsets <math>A, B_1, B_2, \ldots</math> of <math>X,</math><math display=block>\text{if } A \subseteq \bigcup_{j=1}^\infty B_j \text{ then } \mu(A) \leq \sum_{j=1}^\infty \mu(B_j).</math>

Note that there is no subtlety about infinite summation in this definition. Since the summands are all assumed to be nonnegative, the sequence of partial sums could only diverge by increasing without bound. So the infinite sum appearing in the definition will always be a well-defined element of <math>[0, \infty].</math> 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.<ref>The original definition given above follows the widely cited texts of Federer and of Evans and Gariepy. Note that both of these books use non-standard terminology in defining a "measure" to be what is here called an "outer measure."</ref> Some textbooks, such as Halmos (1950) and Folland (1999), instead define an outer measure on <math>X</math> to be a function <math>\mu: 2^X \to [0, \infty]</math> such that

Template:Em: <math>\mu(\varnothing) = 0</math>
Template:Em: if <math>A</math> and <math>B</math> are subsets of <math>X</math> with <math>A \subseteq B,</math> then <math>\mu(A) \leq \mu(B)</math>
for arbitrary subsets <math>B_1, B_2, \ldots</math> of <math>X,</math><math display=block>\mu\left(\bigcup_{j=1}^\infty B_j\right) \leq \sum_{j=1}^\infty \mu(B_j).</math>

Proof of equivalence.

Suppose that <math>\mu</math> is an outer measure in sense originally given above. If <math>A</math> and <math>B</math> are subsets of <math>X</math> with <math>A \subseteq B,</math> then by appealing to the definition with <math>B_1 = B</math> and <math>B_j = \varnothing</math> for all <math>j \geq 2,</math> one finds that <math>\mu(A) \leq \mu(B).</math> The third condition in the alternative definition is immediate from the trivial observation that <math>\cup_j B_j \subseteq \cup_j B_j.</math>

Suppose instead that <math>\mu</math> is an outer measure in the alternative definition. Let <math>A, B_1, B_2, \ldots</math> be arbitrary subsets of <math>X,</math> and suppose that <math display=block>A \subseteq \bigcup_{j=1}^\infty B_j.</math> One then has <math display=block>\mu(A) \leq \mu\left(\bigcup_{j=1}^\infty B_j\right) \leq \sum_{j=1}^\infty\mu(B_j),</math> with the first inequality following from the second condition in the alternative definition, and the second inequality following from the third condition in the alternative definition. So <math>\mu</math> is an outer measure in the sense of the original definition.

Measurability of sets relative to an outer measureEdit

Let <math>X</math> be a set with an outer measure <math>\mu.</math> One says that a subset <math>E</math> of <math>X</math> is <math>\mu</math>-measurable (sometimes called Carathéodory-measurable relative to <math>\mu</math>, after the mathematician Carathéodory) if and only if <math display=block>\mu(A) = \mu(A \cap E) + \mu(A \setminus E)</math> for every subset <math>A</math> of <math>X.</math>

Informally, this says that a <math>\mu</math>-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, for example, should be an outer measure on the plane. One might then expect that every subset of the plane would be deemed "measurable," following the expected principle that <math display=block>\operatorname{area}(A \cup B) = \operatorname{area}(A) + \operatorname{area}(B)</math> whenever <math>A</math> and <math>B</math> 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 measureEdit

It is straightforward to use the above definition of <math>\mu</math>-measurability to see that

if <math>A \subseteq X</math> is <math>\mu</math>-measurable then its complement <math>X \setminus A \subseteq X</math> is also <math>\mu</math>-measurable.

The following condition is known as the "countable additivity of <math>\mu</math> on measurable subsets."

if <math>A_1, A_2, \ldots</math> are <math>\mu</math>-measurable pairwise-disjoint (<math>A_i \cap A_j=\emptyset</math> for <math>i\neq j</math>) subsets of <math>X</math>, then one has <math display=block>\mu\Big(\bigcup_{j=1}^\infty A_j\Big) = \sum_{j=1}^\infty\mu(A_j).</math>

Proof of countable additivity.

One automatically has the conclusion in the form "<math>\,\leq\,</math>" from the definition of outer measure. So it is only necessary to prove the "<math>\,\geq\,</math>" inequality. One has <math display=block>\mu\Big(\bigcup_{j=1}^\infty A_j\Big)\geq\mu\Big(\bigcup_{j=1}^N A_j\Big)</math> for any positive number <math>N,</math> due to the second condition in the "alternative definition" of outer measure given above. Suppose (inductively) that <math display=block>\mu\Big(\bigcup_{j=1}^{N-1} A_j\Big)=\sum_{j=1}^{N-1}\mu(A_j)</math>

Applying the above definition of <math>\mu</math>-measurability with <math>A = A_1 \cup \cdots \cup A_N</math> and with <math>E = A_N,</math> one has <math display=block>\begin{align} \mu\Big(\bigcup_{j=1}^N A_j\Big) &= \mu\left(\Big(\bigcup_{j=1}^N A_j\Big)\cap A_N\right) + \mu\left(\Big(\bigcup_{j=1}^N A_j\Big)\smallsetminus A_N\right) \\ &= \mu(A_N) + \mu\Big(\bigcup_{j=1}^{N-1}A_j\Big) \end{align}</math> which closes the induction. Going back to the first line of the proof, one then has <math display=block>\mu\Big(\bigcup_{j=1}^\infty A_j\Big)\geq\sum_{j=1}^N \mu(A_j)</math> for any positive integer <math>N.</math> One can then send <math>N</math> to infinity to get the required "<math>\,\geq\,</math>" inequality.

A similar proof shows that:

if <math>A_1, A_2, \ldots</math> are <math>\mu</math>-measurable subsets of <math>X,</math> then the union <math>\bigcup_{i=1}^\infty A_i</math> and intersection <math>\bigcap_{i=1}^\infty A_i</math> are also <math>\mu</math>-measurable.

The properties given here can be summarized by the following terminology: Template:Quote One thus has a measure space structure on <math>X,</math> arising naturally from the specification of an outer measure on <math>X.</math> This measure space has the additional property of completeness, which is contained in the following statement:

Every subset <math>A \subseteq X</math> such that <math>\mu(A) = 0</math> is <math>\mu</math>-measurable.

This is easy to prove by using the second property in the "alternative definition" of outer measure.

Restriction and pushforward of an outer measureEdit

Let <math>\mu</math> be an outer measure on the set <math>X </math>.

PushforwardEdit

Given another set <math>Y</math> and a map <math>f:X\to Y </math> define <math>f_\sharp \mu : 2^Y \to [0, \infty]</math> by

<math>\big(f_\sharp\mu\big)(A)=\mu\big(f^{-1}(A)\big).</math>

One can verify directly from the definitions that <math>f_\sharp \mu</math> is an outer measure on <math>Y</math>.

RestrictionEdit

Let Template:Mvar be a subset of Template:Mvar. Define Template:Math by

One can check directly from the definitions that Template:Math is another outer measure on Template:Mvar.

Measurability of sets relative to a pushforward or restrictionEdit

If a subset Template:Mvar of Template:Mvar is Template:Math-measurable, then it is also Template:Math-measurable for any subset Template:Mvar of Template:Mvar.

Given a map Template:Math and a subset Template:Mvar of Template:Mvar, if Template:Math is Template:Math-measurable then Template:Mvar is Template:Math-measurable. More generally, Template:Math is Template:Math-measurable if and only if Template:Mvar is Template:Math-measurable for every subset Template:Mvar of Template:Mvar.

Regular outer measuresEdit

Definition of a regular outer measureEdit

Given a set Template:Mvar, an outer measure Template:Math on Template:Mvar is said to be regular if any subset <math>A\subseteq X</math> can be approximated 'from the outside' by Template:Math-measurable sets. Formally, this is requiring either of the following equivalent conditions:

<math>\mu(A)=\inf\{\mu(B)\mid A\subseteq B, B\text{ is μ-measurable}\}</math>
There exists a Template:Math-measurable subset Template:Mvar of Template:Mvar which contains Template:Mvar and such that <math>\mu(B)=\mu(A)</math>.

It is automatic that the second condition implies the first; the first implies the second by taking the countable intersection of <math>B_i</math> with <math>\mu(B_i)\to\mu(A)</math>

Template:Missing information

The regular outer measure associated to an outer measureEdit

Given an outer measure Template:Math on a set Template:Mvar, define Template:Math by

<math>\nu(A)=\inf\Big\{\mu(B):\mu\text{-measurable subsets }B\subset X\text{ with }B\supset A\Big\}.</math>

Then Template:Math is a regular outer measure on Template:Mvar which assigns the same measure as Template:Math to all Template:Math-measurable subsets of Template:Mvar. Every Template:Math-measurable subset is also Template:Math-measurable, and every Template:Math-measurable subset of finite Template:Math-measure is also Template:Math-measurable.

So the measure space associated to Template:Math may have a larger σ-algebra than the measure space associated to Template:Math. The restrictions of Template:Math and Template:Math 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 Template:Math-measure and finite Template:Math-measure.

From this perspective, Template:Math may be regarded as an extension of Template:Math.

Outer measure and topologyEdit

Suppose Template:Math is a metric space and Template:Math an outer measure on Template:Math. If Template:Math has the property that

<math> \varphi(E \cup F) = \varphi(E) + \varphi(F)</math>

whenever

then Template:Math is called a metric outer measure.

Theorem. If Template:Math is a metric outer measure on Template:Math, then every Borel subset of Template:Math is Template:Math-measurable. (The Borel sets of Template:Math are the elements of the smallest Template:Math-algebra generated by the open sets.)

Construction of outer measuresEdit

Template:See also

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 IEdit

Let Template:Math be a set, Template:Math a family of subsets of Template:Math which contains the empty set and Template:Math a non-negative extended real valued function on Template:Math which vanishes on the empty set.

Theorem. Suppose the family Template:Math and the function Template:Math are as above and define

<math> \varphi(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C\biggr\}.</math>

That is, the infimum extends over all sequences Template:Math of elements of Template:Math which cover Template:Math, with the convention that the infimum is infinite if no such sequence exists. Then Template:Math is an outer measure on Template:Math.

Method IIEdit

The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose Template:Math is a metric space. As above Template:Math is a family of subsets of Template:Math which contains the empty set and Template:Math a non-negative extended real valued function on Template:Math which vanishes on the empty set. For each Template:Math, let

<math>C_\delta= \{A \in C: \operatorname{diam}(A) \leq \delta\} </math>

and

<math> \varphi_\delta(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C_\delta\biggr\}.</math>

Obviously, Template:Math when Template:Math since the infimum is taken over a smaller class as Template:Math decreases. Thus

<math> \lim_{\delta \rightarrow 0} \varphi_\delta(E) = \varphi_0(E) \in [0, \infty]</math>

exists (possibly infinite).

Theorem. Template:Math is a metric outer measure on Template:Math.

This is the construction used in the definition of Hausdorff measures for a metric space.