Outer measure
Encyclopedia
In mathematics
, in particular in 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. A general theory of outer measures was first introduced by Carathéodory to provide a 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
.
Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in R or balls in R3. One might expect to define a generalized measuring function φ on R that fulfils the following requirements:
It turns out that these requirements are incompatible conditions; see non-measurable set
. The purpose of constructing an outer measure on all subsets of X is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
defined on all subsets of X, that satisfies the following conditions:
This allows us to define the concept of measurability as follows: a subset E of X is φ-measurable (or Carathéodory-measurable by φ) iff
for every subset A of X
Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure
.
For a proof of this theorem see the Halmos reference, section 11.
This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure
that is so important for measure theory and the theory of integral
s.
and φ an outer measure on X. If φ has the property that
whenever
then φ is called a metric outer measure
.
Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable. (The Borel set
s of X are the elements of the smallest σ-algebra generated by the open sets.)
Let X be a set, C a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set.
Theorem. Suppose the family C and the function p are as above and define
That is, the infimum
extends over all sequences (Ai)i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.
The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.
Suppose (X,d) is a metric space. As above C is a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set. For each δ > 0, let
and
Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists.
Theorem. φ0 is a metric outer measure on X.
This is the construction used in the definition of Hausdorff measure
s for a metric space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular in measure theory, an outer measure or exterior measure is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by Carathéodory to provide a 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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
(outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem
Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem states that any σ-finite measure defined on a given ring R of subsets of a given set Ω can be uniquely extended to the σ-algebra generated by R...
), and was used in an essential way by Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
to define a dimension-like metric invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
now called Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
.
Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in R or balls in R3. One might expect to define a generalized measuring function φ on R that fulfils the following requirements:
- Any interval of reals [a, b] has measure b − a
- The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
- Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where
) - Countable additivity: for any sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
(Aj) of pairwise disjoint subsets of X
It turns out that these requirements are incompatible conditions; see non-measurable set
Non-measurable set
In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....
. The purpose of constructing an outer measure on all subsets of X is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
Formal definitions
An outer measure on a set X is a functiondefined on all subsets of X, that satisfies the following conditions:
- Null empty set: The empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
has zero outer measure (see also: measure zero).
-
-
- MonotonicityMonotonic functionIn mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
: For any two subsets A and B of X,
- Monotonicity
-
- Countable subadditivity: For any sequence {Aj} of subsets of X (pairwise disjoint or not),
-
-
This allows us to define the concept of measurability as follows: a subset E of X is φ-measurable (or Carathéodory-measurable by φ) iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
for every subset A of X
Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
.
For a proof of this theorem see the Halmos reference, section 11.
This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
that is so important for measure theory and the theory of integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s.
Outer measure and topology
Suppose (X, d) is a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
and φ an outer measure on X. If φ has the property that
whenever
then φ is called a metric outer measure
Metric outer measure
In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space such that\mu = \mu + \mu for every pair of positively separated subsets A and B of X....
.
Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable. (The Borel set
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
s of X are the elements of the smallest σ-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.Let X be a set, C a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set.
Theorem. Suppose the family C and the function p are as above and define
That is, the infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
extends over all sequences (Ai)i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.
The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.
Suppose (X,d) is a metric space. As above C is a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set. For each δ > 0, let
and
Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists.
Theorem. φ0 is a metric outer measure on X.
This is the construction used in the definition of Hausdorff measure
Hausdorff measure
In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...
s for a metric space.