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Measure (mathematics)

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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, integral, and can be generalized to assume signed measure, as with electrical charge. Far-reaching generalizations (such as 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.Archimedes Measuring the 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."Thomas W. Hawkins Jr. (1970) Lebesgue’s Theory of Integration: Its Origins and Development, pages 66,7 University of Wisconsin Press

Definition

Let

X

be a set and

\Sigma

a σ-algebra over

X

, defining subsets of

X

that are "measurable". A set function

\mu

from

\Sigma

to the extended real number line, that is, the real number line together with new (so-called infinite) values

+\infty

and

-\infty

, respectively greater and lower than all other (so-called finite) elements, is called a measure if the following conditions hold:

$\mu(\varnothing) = 0$
Non-negativity: For all $E \in \Sigma, \ \ \mu(E) \geq 0$
Countable additivity (or sigma 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 $\mu(\varnothing) = 0$ is met automatically due to countable additivity: $\mu(E)=\mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing),$ and therefore $\mu(\varnothing)=0.$

Note that any sum involving $+\infty$ will equal $+\infty$ , that is, $a + \infty = +\infty$ for all $a$ in the extended reals.

If the condition of non-negativity is dropped, and $\mu(E)$ only ever equals one of $+\infty$ , $-\infty$ , i.e. no two distinct sets have measures $+\infty$ , $-\infty$ , respectively, then $\mu$ is called a signed measure.

The pair $(X, \Sigma)$ is called a measurable space, and the members of $\Sigma$ are called measurable sets.

A tuple $(X, \Sigma, \mu)$ is called a measure space. A probability measure is a measure with total measure one – that is, $\mu(X) = 1.$ 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 (usually defined on ). 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 Nicolas Bourbaki (2004) and a number of other sources. For more details, see the article on .

Instances

Some important measures are listed here.

The counting measure is defined by $\mu(S)$ = number of elements in $S.$
The Lebesgue measure on $\R$ is a Complete measure translation-invariant measure on a σ-algebra containing the intervals in $\R$ such that $\mu(0,) = 1$ ; and every other measure with these properties extends the Lebesgue measure.
The arc length of interval on the unit circle in the Euclidean plane extends to a measure on the $\sigma$ -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 preserving the circle. Similarly, hyperbolic angle measure is invariant under squeeze mapping.
The Haar measure for a locally compact topological group. For example, $\mathbb R$ is such a group and its Haar measure is the Lebesgue measure; for the unit circle (seen as a subgroup of the multiplicative group of $\mathbb C$ ) its Haar measure is the angle measure. For a discrete group the counting measure is a Haar measure.
Every (pseudo) Riemannian manifold $(M,g)$ has a canonical measure $\mu_g$ that in local coordinates $x_1,\ldots,x_n$ looks like $\sqrt{\left|\det g \right|}d^nx$ where $d^nx$ 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 (and therefore takes all its values in the unit interval 0,). Such a measure is called a probability measure or distribution. See the list of probability distributions for instances.
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 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.

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

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

Countable subadditivity

For any countable sequence

E_1, E_2, E_3, \ldots

of (not necessarily disjoint) measurable sets

E_n

\Sigma:

\mu\left( \bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu(E_i).

Continuity from below

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

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 $n \in \N,$ let $E_n = [n, \infty) \subseteq \R,$ which all have infinite Lebesgue measure, but the intersection is empty.

Other properties

Completeness

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 $\mu(X).$

"Dropping the Edge"

f:X\to0,+\infty

(\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 -\infty,\infty.

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

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|<\infty, 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: $\bigcup_{\alpha\in\lambda} X_\alpha \in \Sigma$ $\mu\left(\bigcup_{\alpha\in\lambda} X_\alpha\right) = \sum_{\alpha\in\lambda}\mu\left(X_\alpha\right).$ The second condition is equivalent to the statement that the ideal of null sets is $\kappa$ -complete.

Sigma-finite measures

A measure space

(X, \Sigma, \mu)

is called finite if

\mu(X)

is a finite real number (rather than

\infty

). 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 with the standard Lebesgue measure are σ-finite but not finite. Consider the $k,$ for all $k;$ there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the 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

X

be a set, let

{\cal A}

be a sigma-algebra on

X,

and let

\mu

be a measure on

{\cal A}.

We say

\mu

is semifinite to mean that for all

A\in\mu^\text{pre}\{+\infty\},

{\cal P}(A)\cap\mu^\text{pre}(\R_{>0})\ne\emptyset.

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.)

Basic examples

Every sigma-finite measure is semifinite.
Assume ${\cal A}={\cal P}(X),$ let $f:X\to0,+\infty,$ and assume $\mu(A)=\sum_{a\in A}f(a)$ for all $A\subseteq X.$

We have that $\mu$ is sigma-finite if and only if $f(x)<+\infty$ for all $x\in X$ and $f^\text{pre}(\R_{>0})$ is countable. We have that $\mu$ is semifinite if and only if $f(x)<+\infty$ for all $x\in X.$
Taking $f=X\times\{1\}$ above (so that $\mu$ is counting measure on ${\cal P}(X)$ ), we see that counting measure on ${\cal P}(X)$ is

sigma-finite if and only if $X$ is countable; and
semifinite (without regard to whether $X$ is countable). (Thus, counting measure, on the power set ${\cal P}(X)$ of an arbitrary uncountable set $X,$ gives an example of a semifinite measure that is not sigma-finite.)

Let $d$ be a complete, separable metric on $X,$ let ${\cal B}$ be the Borel sigma-algebra induced by $d,$ and let $s\in\R_{>0}.$ Then the Hausdorff measure ${\cal H}^s|{\cal B}$ is semifinite.
Let $d$ be a complete, separable metric on $X,$ let ${\cal B}$ be the Borel sigma-algebra induced by $d,$ and let $s\in\R_{>0}.$ Then the packing measure ${\cal H}^s|{\cal B}$ 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

\mu.

It can be shown there is a greatest measure with these two properties:

We say the semifinite part of $\mu$ to mean the semifinite measure $\mu_\text{sf}$ defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:

$\mu_\text{sf}=(\sup\{\mu(B):B\in{\cal P}(A)\cap\mu^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}.$
$\mu_\text{sf}=(\sup\{\mu(A\cap B):B\in\mu^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}\}.$
$\mu_\text{sf}=\mu|_{\mu^\text{pre}(\R_{>0})}\cup\{A\in{\cal A}:\sup\{\mu(B):B\in{\cal P}(A)\}=+\infty\}\times\{+\infty\}\cup\{A\in{\cal A}:\sup\{\mu(B):B\in{\cal P}(A)\}<+\infty\}\times\{0\}.$

Since $\mu_\text{sf}$ is semifinite, it follows that if $\mu=\mu_\text{sf}$ then $\mu$ is semifinite. It is also evident that if $\mu$ is semifinite then $\mu=\mu_\text{sf}.$

Non-examples

Every

0-\infty

measure that is not the zero measure is not semifinite. (Here, we say

0-\infty

measure to mean a measure whose range lies in

\{0,+\infty\}

(\forall A\in{\cal A})(\mu(A)\in\{0,+\infty\}).

) Below we give examples of

0-\infty

measures that are not zero measures.

Let $X$ be nonempty, let ${\cal A}$ be a $\sigma$ -algebra on $X,$ let $f:X\to\{0,+\infty\}$ be not the zero function, and let $\mu=(\sum_{x\in A}f(x))_{A\in{\cal A}}.$ It can be shown that $\mu$ is a measure.

$\mu=\{(\emptyset,0)\}\cup({\cal A}\setminus\{\emptyset\})\times\{+\infty\}.$

$X=\{0\},$ ${\cal A}=\{\emptyset,X\},$ $\mu=\{(\emptyset,0),(X,+\infty)\}.$

Let $X$ be uncountable, let ${\cal A}$ be a $\sigma$ -algebra on $X,$ let ${\cal C}=\{A\in{\cal A}:A\text{ is countable}\}$ be the countable elements of ${\cal A},$ and let $\mu={\cal C}\times\{0\}\cup({\cal A}\setminus{\cal C})\times\{+\infty\}.$ It can be shown that $\mu$ is a measure.

Involved non-example

We say the

\mathbf{0-\infty}

part of

\mu

to mean the measure

\mu_{0-\infty}

defined in the above theorem. Here is an explicit formula for

\mu_{0-\infty}

\mu_{0-\infty}=(\sup\{\mu(B)-\mu_\text{sf}(B):B\in{\cal P}(A)\cap\mu_\text{sf}^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}.

Results regarding semifinite measures

Let $\mathbb F$ be $\R$ or $\C,$ and let $T:L_\mathbb{F}^\infty(\mu)\to\left(L_\mathbb{F}^1(\mu)\right)^*:g\mapsto T_g=\left(\int fgd\mu\right)_{f\in L_\mathbb{F}^1(\mu)}.$ Then $\mu$ is semifinite if and only if $T$ is injective. (This result has import in the study of the dual space of $L^1=L_\mathbb{F}^1(\mu)$ .)
Let $\mathbb F$ be $\R$ or $\C,$ and let ${\cal T}$ be the topology of convergence in measure on $L_\mathbb{F}^0(\mu).$ Then $\mu$ is semifinite if and only if ${\cal T}$ is Hausdorff.
(Johnson) Let $X$ be a set, let ${\cal A}$ be a sigma-algebra on $X,$ let $\mu$ be a measure on ${\cal A},$ let $Y$ be a set, let ${\cal B}$ be a sigma-algebra on $Y,$ and let $\nu$ be a measure on ${\cal B}.$ If $\mu,\nu$ are both not a $0-\infty$ measure, then both $\mu$ and $\nu$ are semifinite if and only if Product measure $(A\times B)=\mu(A)\nu(B)$ for all $A\in{\cal A}$ and $B\in{\cal B}.$ (Here, $\mu\times_\text{cld}\nu$ is the measure defined in Theorem 39.1 in Berberian '65.)

Localizable measures

Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.

Let $X$ be a set, let ${\cal A}$ be a sigma-algebra on $X,$ and let $\mu$ be a measure on ${\cal A}.$

Let $\mathbb F$ be $\R$ or $\C,$ and let $T : L_\mathbb{F}^\infty(\mu) \to \left(L_\mathbb{F}^1(\mu)\right)^* : g \mapsto T_g = \left(\int fgd\mu\right)_{f\in L_\mathbb{F}^1(\mu)}.$ Then $\mu$ is localizable if and only if $T$ is bijective (if and only if $L_\mathbb{F}^\infty(\mu)$ "is" $L_\mathbb{F}^1(\mu)^*$ ).

s-finite measures

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.

Non-measurable sets

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 total variation, hence complex measures include Finite measure 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 , the dual of lp space 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 .

A charge is a generalization in both directions: it is a finitely additive, signed measure.

K. P. S. Bhaskara Rao (1983). 9780120957804, Academic Press. ISBN 9780120957804

(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.)

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