In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and positive operator-valued measures) 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 ÃÂmile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. According to Thomas W. Hawkins Jr., "It was primarily through the theory of multiple integrals and, in particular the work of Camille Jordan that the importance of the notion of measurability was first recognized."
Let be a set and a ÃÂ-algebra over , defining subsets of that are "measurable". A set function from to the extended real number line, that is, the real number line together with new (so-called infinite) values and , respectively greater and lower than all other (so-called finite) elements, is called a measure if the following conditions hold:
If at least one set has finite measure, then the requirement is met automatically due to countable additivity:and therefore
Note that any sum involving will equal , that is, for all in the extended reals.
If the condition of non-negativity is dropped, and only ever equals one of , , i.e. no two distinct sets have measures , , respectively, then is called a signed measure.
The pair is called a measurable space, and the members of are called measurable sets.
A triple is called a measure space. A probability measure is a measure with total measure oneâÂÂâÂÂâÂÂthat is, 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 (usually defined on Hausdorff spaces). When working with locally compact Hausdorff spaces, Radon measures have an alternative, equivalent definition in terms of linear functionals on the locally convex topological vector 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.
Some important measures are listed here.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic 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 for example, 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 be a measure.
If and are measurable sets with then
For any countable sequence of (not necessarily disjoint) measurable sets in
If are measurable sets that are increasing (meaning that ) then the union of the sets is measurable and
If are measurable sets that are decreasing (meaning that ) then the intersection of the sets is measurable; furthermore, if at least one of the has finite measure then
This property is false without the assumption that at least one of the has finite measure. For instance, for each let which all have infinite Lebesgue measure, but the intersection is empty.
A measurable set is called a null set if 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 ÃÂ-algebra of subsets which differ by a negligible set from a measurable set that is, such that the symmetric difference of and is contained in a null set. One defines to equal
If is -measurable, then
for almost all This property is used in connection with Lebesgue integral.
Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set and any set of nonnegative where 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:
The second condition is equivalent to the statement that the ideal of null sets is -complete.
A measure space is called finite if is a finite real number (rather than ). A measure is called ÃÂ-finite if can be decomposed into a countable union of measurable sets of finite measure.
For example, the real numbers with the standard Lebesgue measure are ÃÂ-finite but not finite. The natural numbers are also ÃÂ-finite with respect to counting measure. However, the real numbers are not ÃÂ-finite with respect to counting measure.
Let be a set, let be a sigma-algebra on and let be a measure on We say is semifinite to mean that for all
Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)
The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to It can be shown there is a greatest measure with these two properties:
We say the semifinite part of to mean the semifinite measure defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
Since is semifinite, it follows that if then is semifinite. It is also evident that if is semifinite then
Every measure that is not the zero measure is not semifinite. (Here, we say measure to mean a measure whose range lies in : ) Below we give examples of measures that are not zero measures.
We say the part of to mean the measure defined in the above theorem. Here is an explicit formula for :
Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.
Let be a set, let be a sigma-algebra on and let be a measure on
A measure is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
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 BanachâÂÂTarski 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. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.
Measures that take values in Banach spaces have been studied extensively. 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. More generally see measure theory in topological vector spaces.
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 and the StoneâÂÂÃÂech 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. (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R.)