Measure (mathematics)

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures 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 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, 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."

Definition

Let be a set and a σ-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 values and, respectively greater and lower than all other elements, is called a measure if the following conditions hold:

Non-negativity: For all
Countable additivity : For all countable collections of pairwise disjoint sets in,

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 one – that 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 are Radon measures. 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 and a number of other sources. For more details, see the article on Radon measures.

Instances

Some important measures are listed here.

The counting measure is defined by = number of elements in
The Lebesgue measure on is a complete translation-invariant measure on a σ-algebra containing the intervals in such that ; and every other measure with these properties extends the Lebesgue measure.
The arc length of an interval on the unit circle in the Euclidean plane extends to a measure on the -algebra they generate. It can be called angle measure since the arc length of an interval equals the angle it supports. This measure is invariant under rotations preserving the circle. Similarly, hyperbolic angle measure is invariant under squeeze mapping.
The Haar measure for a locally compact topological group. For example, is such a group and its Haar measure is the Lebesgue measure; for the unit circle its Haar measure is the angle measure. For a discrete group the counting measure is a Haar measure.
Every Riemannian manifold has a canonical measure that in local coordinates looks like where is the usual Lebesgue measure.
The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
Every probability space gives rise to a measure which takes the value 1 on the whole space. Such a measure is called a probability measure or distribution. See the list of probability distributions for instances.
The Dirac measure δ_a is given by δ_a = χ_S, where χ_S is the indicator function of The measure of a set is 1 if it contains the point and 0 otherwise.

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, or another non-negative extensive property, conserved or not. Negative values lead to signed measures, see "generalizations" below.

Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.
Basic properties

Let be a measure.

Monotonicity

If and are measurable sets with then

Measure of countable unions and intersections

Countable subadditivity

For any countable sequence of measurable sets in

Continuity from below

If are measurable sets that are increasing then the union of the sets is measurable and

Continuity from above

If are measurable sets that are decreasing 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.

Other properties

Completeness

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 σ-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

"Dropping the Edge"

If is -measurable, then
for almost all This property is used in connection with Lebesgue integral.

Additivity

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.

Sigma-finite measures

A measure space is called finite if is a finite real number. Nonzero finite measures are analogous to probability measures in the sense that any finite measure is proportional to the probability measure A measure is called σ-finite if 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 for all integers 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'.

Strictly localizable measures

Semifinite measures

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.

Basic examples

Every sigma-finite measure is semifinite.
Assume let and assume for all
* We have that is sigma-finite if and only if for all and is countable. We have that is semifinite if and only if for all
* Taking above, we see that counting measure on is
** sigma-finite if and only if is countable; and
** semifinite.
Let be a complete, separable metric on let be the Borel sigma-algebra induced by and let Then the Hausdorff measure is semifinite.
Let be a complete, separable metric on let be the Borel sigma-algebra induced by and let Then the packing measure is semifinite.
Involved example

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