Measure (mathematics)

For the coalgebraic concept, see Measuring coalgebra.

Not to be confused with Metric (mathematics).

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as 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 projection-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 Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.

Definition

Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

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:

• Non-negativity: For all E in Σ, we have μ(E) ≥ 0.
• Null empty set: $\mu (\varnothing )=0$.
• Countable additivity (or σ-additivity): For all countable collections $\{E_{k}\}_{k=1}^{\infty }$ of pairwise disjoint sets in Σ,
  $$\mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k}).$$

If at least one set $E$ has finite measure, then the requirement that $\mu (\varnothing )=0$ is met automatically. Indeed, by countable additivity,

$\mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),$

and therefore $\mu (\varnothing )=0.$

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.

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.

Instances

Main category: Measures (measure theory)

Some important measures are listed here.

• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure on ℝ is a complete translation-invariant measure on a σ-algebra containing the intervals in ℝ such that μ([0, 1]) = 1; and every other measure with these properties extends Lebesgue measure.
• Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.
• The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
• 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 (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure. See probability axioms.
• The Dirac measure δ_a (cf. Dirac delta function) is given by δ_a(S) = χ_S(a), where χ_S is the indicator function of S. The measure of a set is 1 if it contains the point a 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 (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.

• 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 $\mu$ be a measure.

Monotonicity

If $E_{1}$ and $E_{2}$ are measurable sets with $E_{1}\subseteq E_{2}$ then

$$\mu (E_{1})\leq \mu (E_{2}).$$

Measure of countable unions and intersections

Subadditivity

For any countable sequence $E_{1},E_{2},E_{3},\ldots$ of (not necessarily disjoint) measurable sets $E_{n}$ in $\Sigma :$

$$\mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).$$

Continuity from below

If $E_{1},E_{2},E_{3},\ldots$ are measurable sets that are increasing (meaning that $E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots$) then the union of the sets $E_{n}$ is measurable and

$$\mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).$$

Continuity from above

If $E_{1},E_{2},E_{3},\ldots$ are measurable sets that are decreasing (meaning that $E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots$) then the intersection of the sets $E_{n}$ is measurable; furthermore, if at least one of the $E_{n}$ has finite measure then

$$\mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).$$

This property is false without the assumption that at least one of the $E_{n}$ has finite measure. For instance, for each $n\in \mathbb {N} ,$ let $E_{n}=[n,\infty )\subseteq \mathbb {R} ,$ which all have infinite Lebesgue measure, but the intersection is empty.

Other properties

Completeness

Main article: Complete measure

A measurable set $X$ is called a null set if $\mu (X)=0.$ 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 $\mu (Y)$ to equal $\mu (X).$

μ{x : f(x) ≥ t} = μ{x : f(x) > t} (a.e.)

If $f:X\to [0,+\infty ]$ is $(\Sigma ,{\cal {B}}([0,+\infty ]))$-measurable, then

$$\mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}$$

for almost all $t\in X.$ This property is used in connection with Lebesgue integral.

Proof

Both $F(t):=\mu \{x\in X:f(x)>t\}$ and $G(t):=\mu \{x\in X:f(x)\geq t\}$ are monotonically non-increasing functions of $t,$ so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to the Lebesgue measure. If $t<0$ then $\{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},$ so that $F(t)=G(t),$ as desired.

If $t$ is such that $\mu \{x\in X:f(x)>t\}=+\infty$ then monotonicity implies

$$\mu \{x\in X:f(x)\geq t\}=+\infty ,$$

so that $F(t)=G(t),$ as required. If $\mu \{x\in X:f(x)>t\}=+\infty$ for all $t$ then we are done, so assume otherwise. Then there is a unique $t_{0}\in \{-\infty \}\cup [0,+\infty )$ such that $F$ is infinite to the left of $t$ (which can only happen when $t_{0}\geq 0$) and finite to the right. Arguing as above, $\mu \{x\in X:f(x)\geq t\}=+\infty$ when $t<t_{0}.$ Similarly, if $t_{0}\geq 0$ and $F\left(t_{0}\right)=+\infty$ then $F\left(t_{0}\right)=G\left(t_{0}\right).$

For $t>t_{0},$ let $t_{n}$ be a monotonically non-decreasing sequence converging to $t.$ The monotonically non-increasing sequence $\{x\in X:f(x)>t_{n}\}$ of members of $\Sigma$ has at least one finitely $\mu$-measurable component, and

$$\{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.$$

Continuity from above guarantees that

$$\mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.$$

The right-hand side $\lim _{t_{n}\uparrow t}F\left(t_{n}\right)$ then equals $F(t)=\mu \{x\in X:f(x)>t\}$ if $t$ is a point of continuity of $F.$ Since $F$ is continuous almost everywhere, this completes the proof.

Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set $I$ and any set of nonnegative $r_{i},i\in I$ define:

$$\sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\aleph _{0},J\subseteq I\right\rbrace .$$

That is, we define the sum of the $r_{i}$ to be the supremum of all the sums of finitely many of them.

A measure $\mu$ on $\Sigma$ is $\kappa$-additive if for any $\lambda <\kappa$ and any family of disjoint sets $X_{\alpha },\alpha <\lambda$ the following hold:

$$\bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma$$

$$\mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).$$

Note that the second condition is equivalent to the statement that the ideal of null sets is $\kappa$-complete.

Sigma-finite measures

Main article: Sigma-finite measure

A measure space $(X,\Sigma ,\mu )$ is called finite if $\mu (X)$ is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure $\mu$ is proportional to the probability measure ${\frac {1}{\mu (X)}}\mu$. A measure $\mu$ 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'.

s-finite measures

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.

Non-measurable sets

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.

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. 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.

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. (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.)

See also

• Abelian von Neumann algebra
• Almost everywhere
• Carathéodory's extension theorem
• Content (measure theory)
• Fubini's theorem
• Fatou's lemma
• Fuzzy measure theory
• Geometric measure theory
• Hausdorff measure
• Inner measure
• Lebesgue integration
• Lebesgue measure
• Lorentz space
• Lifting theory
• Measurable cardinal
• Measurable function
• Minkowski content
• Outer measure
• Product measure
• Pushforward measure
• Regular measure
• Vector measure
• Valuation (measure theory)
• Volume form

Bibliography

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External links

• "Measure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Tutorial: Measure Theory for Dummies