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Measure (mathematics)
In mathematics, a measure is a function that assigns a number, e.g., a "size", "volume", or "probability", to subsets of a given set. The concept is important in mathematical analysis and probability theory.
Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and integrals. It is of importance in probability and statistics.
See also Lebesgue integration, Lebesgue measure
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Formal definitions
Formally, a countably additive measure μ is a function defined on a sigma algebra of subsets of X with values in the extended interval [0, ∞] such that the following properties are satisfied:
- The empty set has measure zero:
- Countable additivity: if E_{1}, E_{2}, E_{3}, ... is a sequence of pairwise disjoint sets in ,
The members of are called measurable sets and the structure is called a measure space. The following properties can be derived from the definition above:
- Monotonicity: If E_{1} and E_{2} are measurable sets
- If E_{1}, E_{2}, E_{3}, ... are measurable sets and E_{n} is a subset of E_{n+1} for all n, then the union of the sets E_{n} is measurable
- If E_{1}, E_{2}, E_{3}, ... are measurable sets and E_{n+1} is a subset of E_{n} for all n, then the intersection of the sets E_{n} is measurable; furthermore, if at least one of the E_{n} has finite measure, then
Sigma-finite measures
A measure space Ω is called finite if μ(Ω) is a finite real number (rather than ∞). It is called σ-finite if Ω is the countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very nice properties; σ-finiteness can be compared in this respect to separability of topological spaces.
Completeness
A measurable set S is called a null-set if μ(S) = 0. The measure μ is called complete if every subset of a null-set is measurable (and then automatically itself a null-set).
It is a trivial matter to extend a measure to a complete one; simply consider the σ-algebra of subsets S' which differ by a null set from a measurable set S, that is such that the symmetric difference of S and S' is null.
Examples
Some important measures are listed here.
- The counting measure is defined by μ(S) = number of elements in S.
- The Lebesgue measure is the unique complete translation-invariant measure on a sigma algebra containing the intervals in R such that μ([0,1]) = 1.
- The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
- The zero measure is defined by μ(S) = 0 for all S.
- Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0,1]). Such a measure is called a probability measure. See probability axioms.
Generalizations
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. A measure that takes values in a Banach space is called a spectral measure; these are used mainly in functional analysis for the spectral theorem. To distinguish the usual positive-valued measure from generalizations, we speak of "positive measures".
Another generalization is the finitely additive measure. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limit s, the dual of L^{∞} and the Stone-Čech compactification. All these are linked in one way or another to the axiom of choice.
The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in R^{n} consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k=0,1,2,...,n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c>0 multiplies the set's "measure" by c^{k}. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n-1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogenous of degree 0 is the Euler characteristic.
See also
References
- P. Halmos, Measure theory, D. van Nostrand and Co., 1950
- M. E. Munroe, Introduction to Measure and Integration, Addison Wesley, 1953