Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after some adjustments — electrical charge. These seemingly distinct concepts are innately very similar and may, in many cases, be treated as mathematically indistinguishable. Measures are foundational in probability theory. Far-reaching generalizations of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
If at least one set has finite measure, then the requirement that is met automatically. Indeed, by countable additivity,
and therefore
If the condition of non-negativity is omitted but the second and third of these conditions are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.
The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces, then a function is called measurable if for every Y-measurable set , the inverse image is X-measurable – i.e.: . In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See also Measurable function § Term usage variations about another setup.
A triple (X, Σ, μ) is called a measure space. A probability measure is a measure with total measure one – i.e. μ(X) = 1. A probability space is a measure space with a probability measure.
For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.
Main category: Measures (measure theory) |
Some important measures are listed here.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure.
In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.
Let μ be a measure.
If E_{1} and E_{2} are measurable sets with E_{1} ⊆ E_{2} then
For any countable sequence E_{1}, E_{2}, E_{3}, ... of (not necessarily disjoint) measurable sets E_{n} in Σ:
If E_{1}, E_{2}, E_{3}, ... are measurable sets and for all n, then the union of the sets E_{n} is measurable, and
If E_{1}, E_{2}, E_{3}, ... are measurable sets and, 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
This property is false without the assumption that at least one of the E_{n} has finite measure. For instance, for each n ∈ N, let E_{n} = [n, ∞) ⊂ R, which all have infinite Lebesgue measure, but the intersection is empty.
Main article: Complete measure |
A measurable set X is called a null set if μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.
A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).
If the -measurable function takes values on then
for almost all with respect to the Lebesgue measure.^{[1]} This property is used in connection with Lebesgue integral.
Proof. |
Both and are monotonically non-increasing functions of so both of them are continuous almost everywhere, relative to the Lebesgue measure. If for all , then, by additivity and non-negativity, as required. If, on the contrary, for some then there is a unique such that this function is infinite to the left of (which can only happen when and finite to the right. Arguing as above, when For let be a monotonically non-decreasing sequence converging to The monotonically non-increasing sequence of -measurable sets has at least one finitely -measurable element, and Continuity from above shows that The right-hand side then equals if is a point of continuity. |
Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set and any set of nonnegative define:
That is, we define the sum of the to be the supremum of all the sums of finitely many of them.
A measure on is -additive if for any and any family of disjoint sets the following hold:
Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.
Main article: Sigma-finite measure |
A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure . A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a 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 space 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 convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
Main article: s-finite measure |
A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
Main article: Non-measurable set |
If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.
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. Measures that take values in Banach spaces have been studied extensively.^{[2]} A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures.
Another generalization is the finitely additive measure, also known as a content. 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. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L^{∞} and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.
A charge is a generalization in both directions: it is a finitely additive, signed measure.