Measure

measure of a set

A notion that generalizes those of the length of segments, the area of figures and the volume of bodies, and that corresponds intuitively to the mass of a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection with the study and improvement of the notion of an integral.

Definition and general properties.[edit]

Let $X$ be a set and let $\mathcal{E}$ be a class of subsets of $X$. A non-negative (not necessarily finite) set function $\lambda$ defined on $\mathcal{E}$ is called additive, finitely additive or countably additive if

$$\lambda \left(\bigcup _{i=1}^n{E}_i\right)=\sum _{i=1}^n\lambda (E_i)$$

whenever

$$E_i\in \mathcal{E},\bigcup _{i=1}^n{E}_i\in \mathcal{E},E_i\bigcap E_j=\mathrm{\varnothing },i\ne j,$$

for, respectively, $n=2$, $n$ arbitrary finite, and $n\le \mathrm{\infty }$.

A collection $\mathcal{P}$ of subsets of $X$ is called a semi-ring of sets if

1. $\mathrm{\varnothing }\in \mathcal{P}$
2. $E_1,E_2\in \mathcal{P}⟹E_1\cap E_2\in \mathcal{P}$
3. $E,E_1\in \mathcal{P},E_1\subset E⟹E$ is representable as $E=\bigcup _{i=1}^n{E}_i,E_i\cap E_j=\mathrm{\varnothing }$ for $i\ne j,E_i\in \mathcal{P},i=1\dots n,n<\mathrm{\infty }$ .

A collection $\mathcal{R}$ of subsets of $X$ is called a ring of sets if

1. $\mathrm{\varnothing }\in \mathcal{R}$
2. $E_1,E_2\in \mathcal{R}⟹E_1\cup E_2\in \mathcal{R},E_1\setminus E_2\in \mathcal{R}$.

An example of a semi-ring is: $X={\mathbf{R}}^k$, $\mathcal{P}$ is the collection of all intervals of the form

$$\{x=(x_1,\dots ,x_k)\in {\mathbf{R}}^k\mid a_i\le x_i<b_i,i=1,\dots ,k\}$$

where $a_i,b_i\in \mathbf{R}$ for $i=1,\dots ,k$. The collection of all possible finite unions of such intervals is a ring.

A collection $\mathcal{S}$ of subsets of $X$ is called a $\sigma$-ring if

1. $\mathrm{\varnothing }\in \mathcal{S}$
2. $E_1,E_2\in \mathcal{S}⟹E_1\setminus E_2\in \mathcal{S}$
3. $E_i\in \mathcal{S}(i=1,2,\dots )⟹\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\in S$.

Every $\sigma$-ring is a ring; every ring is a semi-ring.

A finitely-additive measure is a non-negative finitely-additive set function $m$ such that $m(\mathrm{\varnothing })=0$. The domain of definition ${\mathcal{E}}_m$ of a finitely-additive measure may be a semi-ring, a ring or a $\sigma$-ring. In the definition of a finitely-additive measure on a ring or on a $\sigma$-ring the condition of finite additivity can be weakened to additivity, which leads to the same notion.

If $m$ is a finitely-additive measure, if the sets $E,E_1,\dots ,E_n$ belong to its domain of definition, and if $E\subset \bigcup _{i=1}^n{E}_i$ , then

$$m(E)\le \sum _{i=1}^{n}m(E_i).$$

Let $m_1$ be a finitely-additive measure with domain ${\mathcal{E}}_{m_1}$. A finitely-additive measure $m_2$ with domain ${\mathcal{E}}_{m_2}$ is called an extension of $m_1$ if ${\mathcal{E}}_{m_1}\subset {\mathcal{E}}_{m_2}$ and $m_2(E)=m_1(E)\mathrm{\forall }E\in {\mathcal{E}}_{m_1}$.

Every finitely-additive measure $m$ defined on a semi-ring $\mathcal{P}$ admits a unique extension to a finitely-additive measure $m^{\prime }$ on the smallest ring $\mathcal{R}(\mathcal{P})$ containing $\mathcal{P}$. This extension is defined as follows: Every $E\in \mathcal{R}(\mathcal{P})$ is representable as $E=\bigcup _{i=1}^n{E}_i,E_i\in \mathcal{P},E_i\cap E_j=\mathrm{\varnothing },i\ne j$, and one sets

$$m^{\prime }(E)=\sum _{i=1}^{n}m(E_i).$$

A finitely-additive measure that has the property of countable additivity is called a measure. Examples of measures: Let $X$ be an arbitrary non-empty set, let ${\mathcal{E}}_{\mu }$ be a $\sigma$-ring, a ring or a semi-ring of subsets of $X$, let $\{x_1,x_2,\dots \}$ be a countable subset of $X$, and let $p_1,p_2,\dots$ be non-negative numbers. Then the function

$$\mu (E)=\sum _n{p}_n{\delta }_{x_n}(E),$$ where ${\delta }_x(E)=1$ if $x\in E$ and ${\delta }_x(E)=0$ if $x\notin E$, is a measure defined on ${\mathcal{E}}_{\mu }$. The measures ${\delta }_x$ are called elementary, degenerate or Dirac measures (sometimes, Dirac masses). Not every finitely-additive measure is a measure. For example, if $X$ is the set of rational points of the segment $[0,1]$, $\mathcal{P}$ is the semi-ring of all possible intersections of subintervals of $[0,1]$ with $X$, and for every $a,b$, $0\le a\le b\le 1$,

$$m((a,b)\cap X)=m([a,b)\cap X)=m((a,b]\cap X)=m([a,b]\cap X)=b-a,$$

then $m$ is finitely additive, but not countably additive on $\mathcal{P}$.

A (finitely-additive) measure $m$ with domain ${\mathcal{E}}_m$ is said to be finite (respectively, $\sigma$-finite) if $m(E)<\mathrm{\infty }$ for all $E\in {\mathcal{E}}_m$ (respectively, if for every $E\in {\mathcal{E}}_m$ there is a sequence of sets $\{E_i\}$ in ${\mathcal{E}}_m$ such that $E\subset \bigcup _{i=1}^{\mathrm{\infty }}{E}_i$ and $m(E_i)<\mathrm{\infty }$, $i=1,2,\dots$).

A (finitely-additive) measure $m$ is said to be totally finite (totally $\sigma$-finite) if it is finite (respectively, $\sigma$-finite) and $X\in {\mathcal{E}}_m$.

A pair $(X,\mathcal{S})$, where $X$ is a set and $\mathcal{S}$ is a $\sigma$-ring of subsets of $X$ such that $\bigcup _{E\in \mathcal{S}}E=X$, is called a measurable space. A triple $(X,\mathcal{S},\mu )$, where $(X,\mathcal{S})$ is a measurable space and $\mu$ is a measure on $\mathcal{S}$, is called a measure space. A space with a totally-finite measure $\mu$ normalized by the condition $\mu (X)=1$ is called a probability space. In abstract measure theory, where the basic notions are a measurable space $(X,\mathcal{S})$ or a measure space $(X,\mathcal{S},\mu )$, the elements of $\mathcal{S}$ are also referred to as measurable sets (cf. also Measurable set).

Properties of measure spaces.[edit]

Let $\{E_i\}$ be an arbitrary sequence of measurable sets. Then

1. $\mu (lim\underset{i\to \mathrm{\infty }}{inf}{E}_i)\le lim\underset{i\to \mathrm{\infty }}{inf}\mu (E_i)$
2. if $\mu (\bigcup _{i=i_0}^{\mathrm{\infty }}{E}_i)<\mathrm{\infty }$ for some $i_0$, then $$\mu \left(\underset{i\to \mathrm{\infty }}{lim sup}{E}_i\right)\ge \underset{i\to \mathrm{\infty }}{lim sup}\mu (E_i)$$
3. if $\underset{i\to \mathrm{\infty }}{lim}{E}_i$ exists and the condition in 2) is satisfied, then

$$\mu \left(\underset{i\to \mathrm{\infty }}{lim}{E}_i\right)=\underset{i\to \mathrm{\infty }}{lim}\mu (E_i)$$

A finitely-additive measure $m$ defined on a ring $\mathcal{R}$ is a measure if and only if

$$m\left(\underset{i\to \mathrm{\infty }}{lim}{E}_i\right)=\underset{i\to \mathrm{\infty }}{lim}m(E_i)$$

for every monotone increasing sequence $\{E_i\}$ of elements of $\mathcal{R}$ such that $\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\in \mathcal{R}$.

Let $(X_1,{\mathcal{S}}_1,{\mu }_1)$ be a measure space, let $(X_2,{\mathcal{S}}_2)$ be a measurable space and let $T$ be a measurable mapping from $X_1$ into $X_2$, i.e.

$$T^{-1}(E)=\{x\in X_1:Tx\in E\}\in {\mathcal{S}}_1$$

for all $E\in {\mathcal{S}}_2$. The measure generated by the mapping $T$ (denoted here by $\mu T^{-1}$) is the measure on ${\mathcal{S}}_2$ defined by

$$\mu T^{-1}(E)=\mu (T^{-1}E).$$

Let $(X,\mathcal{S},\mu )$ be a measure space and let $X_1\subset X$. Define ${\mu }_{X_1}$ on the sets $E$ from the $\sigma$-ring $\mathcal{S}\cap X_1=\{E\cap X_1:E\in {\mathcal{S}}_1\}$ by

$${\mu }_{X_1}(E)=\underset{E\subset F\in \mathcal{S}}{inf}\mu (F).$$

Then $(X_1,\mathcal{S}\cap X_1,{\mu }_{X_1})$ is a measure space; ${\mu }_{X_1}$ is called the restriction of the measure $\mu$ to $X_1$.

An atom of the space $(X,\mathcal{S},\mu )$ (or of the measure $\mu$) is any set $E\in \mathcal{S}$ of positive measure such that if $F\subset E$, $F\in \mathcal{S}$, then either $\mu (F)=0$ or $\mu (F)=\mu (E)$. A measure space without atoms is called non-atomic or continuous (in this case $\mu$ is also called non-atomic or continuous). If $(X,\mathcal{S},\mu )$ is a space with a non-atomic $\sigma$-finite measure and $E_1\in \mathcal{S}$, then for every $\alpha$ with $0\le \alpha \le \mu (E_1)$ (possibly $\alpha =\mathrm{\infty }$) there is an element $E_2\in \mathcal{S}$ such that $E_2\subset E_1$ and $\mu (E_2)=\alpha$.

A measure space $(X,\mathcal{S},{\mu }_1)$ (or the measure $\mu$) is said to be complete if $E\in \mathcal{S}$, $F\subset E$, $\mu (E)=0$ imply $F\in \mathcal{S}$. Every measure space $(X,\mathcal{S},\mu )$ can be completed by adjoining to $\mathcal{S}$ all the sets of the form $E\cup N$ with $E\in \mathcal{S}$, $N\subset N^{\prime }$, $N^{\prime }\in S$, $\mu (N^{\prime })=0$, and putting for such sets $\bar{\mu }(E\cup N)=\mu (E)$. The class of sets of the indicated form is a $\sigma$-ring, and $\bar{\mu }$ is a complete measure on it. The sets of null measure are called null sets. If the set of points of $X$ at which a property $Q$ is not satisfied is a null set, then property $Q$ is said to hold [[Almost- everywhere|almost-everywhere]].

Extension of measures.[edit]

A measure ${\mu }_2$ is an extension of a measure ${\mu }_1$ if ${\mu }_2$ is an extension of ${\mu }_1$ in the class of finitely-additive measures (see above). Every measure defined on a semi-ring $\mathcal{P}$ admits a unique extension to a measure on the ring $\mathcal{R}(\mathcal{P})$ generated by $\mathcal{P}$( the extension is realized in the same way as in the case of finitely-additive measures). Further, every measure $\mu$ defined on a ring $\mathcal{R}$ can be extended to a measure ${\mu }^{\prime }$ on the $\sigma$- ring $\mathcal{S}(\mathcal{R})$ generated by $\mathcal{R}$; if $\mu$ is $\sigma$- finite, then ${\mu }^{\prime }$ is unique and $\sigma$- finite. The value of ${\mu }^{\prime }$ on any set $E\in \mathcal{S}(\mathcal{R})$ can be given by the formula

$$\begin{array}{}\text{(*)}& {\mu }^{\prime }(E)=\inf \{\sum _{i=1}^{\mathrm{\infty }}\mu (E_i):E_i\in \mathcal{R},i=1,2\dots E\subset {\cup }_{i=1}^{\mathrm{\infty }}{E}_i\}.\end{array}$$

A class of subsets of $X$ is called hereditary if it contains, together with any set in the class, all its subsets. An outer measure is a set function $m^{\star }$, defined on a hereditary $\sigma$- ring $\mathcal{H}$( i.e. a class of sets which is simultaneously hereditary and a $\sigma$- ring), which has the following properties:

1) $0\le m^{\star }(E)\le \mathrm{\infty }$, $m^{\star }(\mathrm{\varnothing })=0$;

2) $E\subset F$ implies $m^{\star }(E)\le m^{\star }(F)$;

3) $m^{\star }({\cup }_i=1^{\mathrm{\infty }}{E}_i)\le \sum _i=1^{\mathrm{\infty }}{m}^{\star }(E_i)$.

Given a measure $\mu$ on the ring $\mathcal{R}$ one can construct an outer measure ${\mu }^{\star }$ on the hereditary $\sigma$- ring $\mathcal{H}(\mathcal{R})$ generated by $\mathcal{R}$( $\mathcal{H}(\mathcal{R})$ consists of all sets that can be covered by a countable union of elements of $\mathcal{R}$) by means of the formula

$${\mu }^{\star }(E)=\inf \{\sum _{i=1}^{\mathrm{\infty }}\mu (E_i):E_i\in \mathcal{R},i=1,2\dots E\subset {\cup }_{i=1}^{\mathrm{\infty }}{E}_i\}.$$

The outer measure ${\mu }^{\star }$ is called the outer measure induced by the measure $\mu$.

Let $m^{\star }$ be an outer measure on a hereditary $\sigma$- ring $\mathcal{H}$ of subsets of $X$. A set $E\in \mathcal{H}$ is called $m^{\star }$- measurable if

$$m^{\star }(A)=m^{\star }(A\cap E)+m^{\star }(A\cap (X\setminus E))$$

for every $A\in \mathcal{H}$. The collection $\bar{\mathcal{S}}$ of $m^{\star }$- measurable sets is a $\sigma$- ring which contains all sets of null outer measure. The set function $\bar{m}$ on $\bar{\mathcal{S}}$ defined by the equality $\bar{m}(E)=m^{\star }(E)$ is a complete measure and is called the measure induced by the outer measure $m^{\star }$.

Suppose that $\mu$ is a measure on a ring $\mathcal{R}$ and that ${\mu }^{\star }$ is the outer measure on $\mathcal{H}(\mathcal{R})$ induced by $\mu$. Let $\bar{\mathcal{S}}$ and $\bar{\mu }$ denote the collection of ${\mu }^{\star }$- measurable sets and the measure on $\bar{\mathcal{S}}$ induced by ${\mu }^{\star }$, respectively. Then $\bar{\mu }$ is an extension of $\mu$, and since $\mathcal{S}(\mathcal{R})\subset \bar{\mathcal{S}}$ it follows that the function ${\mu }^{\prime }$ on $\mathcal{S}(\mathcal{R})$ given by formula (*) is also a measure extending $\mu$. If the original measure $\mu$ on $\mathcal{R}$ is $\sigma$- finite, then the space $(X,\bar{\mathcal{S}},\bar{\mu })$ is the completion of the space $(X,\mathcal{S}(\mathcal{R}),{\mu }^{\prime })$( see (*)). If $\mu$ is given on the $\sigma$- ring $\mathcal{S}$, then the induced outer measure ${\mu }^{\star }$ on the hereditary $\sigma$- ring $\mathcal{H}(\mathcal{S})$ generated by $\mathcal{S}$ is given by the formula

$${\mu }^{\star }(E)=\inf \{\mu (F):E\subset F,F\in \mathcal{S}\}.$$

Alongside with the outer measure ${\mu }^{\star }$, one defines the inner measure induced by the measure $\mu$ on $\mathcal{S}$. It is defined as

$${\mu }_{\star }(E)=sup\{\mu (F):E\supset F,F\in \mathcal{S}\},E\in \mathcal{H}(\mathcal{S}).$$

For every set $E\in \mathcal{H}(\mathcal{S})$ a measurable kernel $E^{\prime }$ and a measurable envelope $E^{\prime \prime }$ are defined as elements of $\mathcal{S}$ such that $E^{\prime }\subset E\subset E^{\prime \prime }$ and $\mu (F^{\prime })=\mu (F^{\prime \prime })=0$ for all $F^{\prime },F^{\prime \prime }\in \mathcal{S}$ such that $F^{\prime }\subset E\setminus E^{\prime }$, $F^{\prime \prime }\subset E^{\prime \prime }\setminus E$. A measurable kernel exists always, while a measurable envelope exists whenever $E$ has $\sigma$- finite outer measure; moreover, ${\mu }_{\star }(E)=\mu (E^{\prime })$ and ${\mu }^{\star }(E)=\mu (E^{\prime \prime })$. Let $\mu$ be a measure on a ring $\mathcal{R}$ and let ${\mu }^{\prime }$ be its extension to the $\sigma$- ring $\mathcal{S}(\mathcal{R})$ generated by $\mathcal{R}$. The inner measure ${\mu }_{\star }^{\prime }$ on the subsets $E$ of finite $\mu$- measure can be expressed in terms of the outer measure ${\mu }^{\star }$( and hence $\mu$):

$${\mu }_{\star }^{\prime }(A)=\mu (E)-{\mu }^{\star }(E\setminus A),A\subset E.$$

Furthermore, a set $F$ belonging to the hereditary $\sigma$- ring $\mathcal{H}(\mathcal{R})$ with finite outer ${\mu }^{\star }$- measure is ${\mu }^{\star }$- measurable if and only if ${\mu }^{\star }(F)={\mu }_{\star }^{\prime }(F)$. In case the original measure $\mu$ on $\mathcal{R}$ is totally finite, one has the following necessary and sufficient condition for the ${\mu }^{\star }$- measurability of a set $E\subset X$:

$$\mu (X)={\mu }^{\star }(E)+{\mu }^{\star }(X\setminus E).$$

For totally-finite measures on $\mathcal{R}$ this condition is frequently taken as the definition of ${\mu }^{\star }$- measurability of the set $E$.

If $(X,\mathcal{S},\mu )$ is a space with a $\sigma$- finite measure and $X_1\dots X_n$ is a finite collection of elements of the hereditary $\sigma$- ring $\mathcal{H}(\mathcal{S})$ generated by $\mathcal{S}$, then on the $\sigma$- ring $\tilde{\mathcal{S}}$ generated by $\mathcal{S}$ and the sets $X_1\dots X_n$ one can define a measure $\tilde{\mu }$ which agrees with $\mu$ on $\mathcal{S}$.

Jordan, Lebesgue and Lebesgue–Stieltjes measures.[edit]

An example of an extension of a measure is provided by the Lebesgue measure in ${\mathbf{R}}^k$. The intervals of the form

$$I=\{(x_1\dots x_k):a_i\le x_i<b_i,i=1\dots k\}$$

form a semi-ring $\mathcal{P}$ in ${\mathbf{R}}^k$. For each such interval, let

$$\lambda (I)=\prod _{i=1}^k(b_i-a_i)$$

( $\lambda (I)$ coincides with the volume of $I$). The function $\lambda$ is $\sigma$- finite and countably additive on $\mathcal{P}$ and admits a unique extension to a measure ${\lambda }^{\prime }$ on the $\sigma$- ring $\mathcal{S}$ generated by $\mathcal{P}$; $\mathcal{S}$ is identical with the $\sigma$- ring of Borel sets (cf. Borel set) (or Borel-measurable sets) in ${\mathbf{R}}^k$. The measure ${\lambda }^{\prime }$ was first defined by E. Borel in 1898 (see Borel measure). The completion $\bar{\lambda }$ of ${\lambda }^{\prime }$( defined on $\bar{\mathcal{S}}$) is called the Lebesgue measure, and was introduced by H. Lebesgue in 1902 (see Lebesgue measure). A set belonging to the domain $\bar{\mathcal{S}}$ of $\bar{\lambda }$ is called Lebesgue measurable. A bounded set $E\subset {\mathbf{R}}^k$ belongs to $\bar{\mathcal{S}}$ if and only if $\lambda (I)={\lambda }^{\star }(E)+{\lambda }^{\star }(I\setminus E)$, where $I\in \mathcal{P}$ is some interval containing $E$; in this case $\bar{\lambda }(E)={\lambda }^{\star }(E)$. A set $E\subset {\mathbf{R}}^k$ belongs to $\bar{\mathcal{S}}$ if and only if for some sequence $\{r_n\}$, $r_n>0$, $n=1,2\dots$ such that $r_n\to \mathrm{\infty }$, one has $E\cap B_{r_n}\in \bar{\mathcal{S}}$ for all $n$, where $B_r=\{x\in {\mathbf{R}}^k:\Vert x\Vert \le r\}$. The cardinality of the family of all Borel sets in ${\mathbf{R}}^k$ is $\mathfrak{c}$( the cardinality of the continuum), whereas the cardinality of the family of all Lebesgue-measurable sets is $2^{\mathfrak{c}}$, so that the inclusion $\mathcal{S}\subset \bar{\mathcal{S}}$ is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.

The Lebesgue measure $\bar{\lambda }$ is invariant under linear orthogonal transformations $A$ of ${\mathbf{R}}^k$ as well as under translations by elements $x\in {\mathbf{R}}^k$, i.e. $\bar{\lambda }(AE+x)=\bar{\lambda }(E)$ for all $E\in \mathcal{S}$.

Using the axiom of choice one can show that there exist sets which are not Lebesgue measurable. On the straight line, for example, such a set can be obtained by picking one point in each coset in $\mathbf{R}$ of the additive subgroup of rational numbers (Vitali's example).

Historically the Borel and Lebesgue measures in ${\mathbf{R}}^k$ were preceded by the measure defined by C. Jordan in 1892 (see Jordan measure). The idea of the definition of the Jordan measure is very close to that of the classic definition of area and volume, which goes back to ancient Greece. Thus, a set $E\subset {\mathbf{R}}^k$ is called Jordan measurable if there exist two sets, representable as finite unions of disjoint rectangles, one contained in $E$ and the other containing $E$, such that the difference of their volumes (defined in an obvious manner) is arbitrarily small. The Jordan measure of such a set is the infimum of the volumes of finite unions of rectangles covering $E$. A Jordan-measurable set is also Lebesgue measurable, and its Jordan and Lebesgue measures are equal. The domain of the Jordan measure is merely a ring, and not a $\sigma$- ring, which restricts considerably its domain of applicability.

The Lebesgue measure is a particular case of the more general Lebesgue–Stieltjes measure. The latter is defined by means of a real-valued function $F$ on ${\mathbf{R}}^k$ with the properties:

1) $-\mathrm{\infty }<F<\mathrm{\infty }$;

2) ${\mathrm{\Delta }}_{b_1-a_1}\dots {\mathrm{\Delta }}_{b_k-a_k}F(a_1\dots a_k)\ge 0$ for $a_i<b_i$, $i=1\dots k$, where ${\mathrm{\Delta }}_{b_i-a_i}$ is the difference operator with step $b_i-a_i$ taken at the point $a_i$ with respect to the $i$- th coordinate;

3) $F(a_1\dots a_k)\uparrow F(b_1\dots b_k)$ as $a_i\uparrow b_i$, $i=1\dots k$.

Given such a function $F$, the measure ${\mu }_F$ of the interval

$$I=\{(x_1\dots x_k):a_i\le x_i<b_i,i=1\dots k\}$$

is defined by the formula

$${\mu }_F(I)={\mathrm{\Delta }}_{b_1-a_1}\dots {\mathrm{\Delta }}_{b_k-a_k}F(a_1\dots a_k).$$

It turns out that ${\mu }_F$ is countably additive on the semi-ring of all such intervals and that it admits an extension to the $\sigma$- algebra of Borel sets; the completion of this extension yields what is called the Lebesgue–Stieltjes measure corresponding to $F$. For the particular choice

$$F(x_1\dots x_k)=x_1\dots x_k$$

one obtains the Lebesgue measure.

Measures in product spaces.[edit]

By definition, the product of two measurable spaces $(X_1,{\mathcal{S}}_1)$, $(X_2,{\mathcal{S}}_2)$ is the measurable space consisting of the set $X_1\times X_2=\{(x_1,x_2):x_1\in X_1,x_2\in X_2\}$( the product of $X_1$ and $X_2$) and the $\sigma$- ring ${\mathcal{S}}_1\times {\mathcal{S}}_2$ of subsets of $X$( the product of the $\sigma$- rings ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$) generated by the semi-ring $\mathcal{P}$ of sets of the form

$$E_1\times E_2=\{(x_1,x_2):x_1\in E_1,x_2\in E_2\},$$

where $E_1,E_2\in \mathcal{S}$. If $(X_1,{\mathcal{S}}_1,{\mu }_1)$ and $(X_2,{\mathcal{S}}_2,{\mu }_s)$ are measure spaces, the formula

$$\mu (E_1\times E_2)={\mu }_1(E_1){\mu }_2(E_2),E_1\in {\mathcal{S}}_1,E_2\in {\mathcal{S}}_2,$$

defines a measure on $\mathcal{P}$; if ${\mu }_1$ and ${\mu }_2$ are $\sigma$- finite, $\mu$ extends uniquely to a measure on ${\mathcal{S}}_1\times {\mathcal{S}}_2$, denoted by ${\mu }_1\times {\mu }_2$. The measure ${\mu }_1\times {\mu }_2$ and the space $(X_1\times X_2,{\mathcal{S}}_1\times {\mathcal{S}}_2,{\mu }_1\times {\mu }_2)$ are called, respectively, the product of the measures ${\mu }_1$ and ${\mu }_2$, and the product of the measure spaces $(X_1,{\mathcal{S}}_1,{\mu }_1)$ and $(X_2,{\mathcal{S}}_2,{\mu }_2)$. The completion of the product of the Lebesgue measure in ${\mathbf{R}}^k$ and the Lebesgue measure in ${\mathbf{R}}^l$ is the Lebesgue measure in ${\mathbf{R}}^{k+l}$. Analogously one defines the product of an arbitrary finite number of measure spaces.

Let $(X_i,{\mathcal{S}}_i,{\mu }_i)$, $i\in I$, be an arbitrary family of measure spaces such that ${\mu }_i(X_i)=1$, $i\in I$. The product space $X=\prod _{i\in I}{X}_i$ is, by definition, the set of all functions on $I$ such that the value at each $i\in I$ is an element $x_i\in X_i$. A measurable rectangle in $X$ is any set of the form $\prod _{i\in I}{E}_i$, where $E_i\in {\mathcal{S}}_i$ and only finitely many sets $E_i$ are different from $X_i$. The family of measurable rectangles forms a semi-ring $\mathcal{P}$. The $\sigma$- ring generated by $\mathcal{P}$ is denoted by $\prod _{i\in I}{\mathcal{S}}_i$ and is called the product of the $\sigma$- rings ${\mathcal{S}}_i$. Now, let $\mu$ be the function on $\mathcal{P}$ defined by $\mu (E)=\prod _{i\in I}{\mu }_i(E_i)$ for $E=\prod _{i\in I}{E}_i$. The function $\mu$ thus defined is a measure which admits a unique extension to a measure on $\prod _{i\in I}{\mathcal{S}}_i$, denoted by $\prod _{i\in I}{\mu }_i$. The measure space $(\prod _{i\in I}{X}_i,\prod _{i\in I}{\mathcal{S}}_i,\prod _{i\in I}{\mu }_i)$ is called the product of the spaces $(X_i,{\mathcal{S}}_i,{\mu }_i)$, $i\in I$.

The product of an arbitrary number of measure spaces is a particular case of the following, more general, scheme, which plays an important role in probability theory. Let $(X_i,{\mathcal{S}}_i)$, $i\in I$, be a family of measurable spaces (each ${\mathcal{S}}_i$ is a $\sigma$- algebra), and suppose that for each finite subset $I_1\subset I$ there is given a probability measure ${\mu }_{I_1}$ on the measurable spaces $(\prod _{i\in I_1}{X}_i,\prod _{i\in I_1}\in {\mathcal{S}}_i)$( the product of measures corresponds to the case that ${\mu }_{I_1}=\prod _{i\in I_1}{\mu }_i$ for all finite $I_1\subset I$). Suppose further that each two measures ${\mu }_{I_1},{\mu }_{I_2}$ are compatible in the sense that if $I_1\subset I_2$ and $p_{21}$ is the projection of $\prod _{i\in I_2}{X}_i$ onto $\prod _{i\in I_1}{X}_i$, then ${\mu }_{I_1}(E)={\mu }_{I_2}{p}_2{1}^{-1}(E)$ for all $E\in \prod _{i\in I_1}{\mathcal{S}}_i$( by definition, $p_{21}$ is the mapping of $\prod _{i\in I_2}{X}_i$ onto $\prod _{i\in I_1}{X}_i$ such that $(p_{21}(x){)}_i=x_i$ for all $i\in I_1$). The following question arises: Is there a probability measure on $\prod _{i\in I}{\mathcal{S}}_i$ such that ${\mu }_{I_1}(E)=\mu p^{-1}(E)$ for every finite $I_1\subset I$ and every $E\in \prod _{i\in I_1}{\mathcal{S}}_i$, where $p$ denotes the projection of $\prod _{i\in I}{X}_i$ onto $\prod _{i\in I_1}{X}_i$? It turns out that such a measure does not always exist, and additional conditions must be imposed to guarantee its existence. One such condition is perfectness of the measures ${\mu }_i$( corresponding to the one-point sets $i\in I$). The notion of a perfect measure was first introduced by B.V. Gnedenko and A.N. Kolmogorov [6]. A space $(X,\mathcal{S},\mu )$ with a totally-finite measure, as well as the measure $\mu$ itself, is called perfect if for every $\mathcal{S}$- measurable real-valued function $f$ on $X$ there is a Borel set $B\subset f(X)$ such that $\mu (f^{-1}(B))=\mu (X)$. The perfectness assumption eliminates a series of "pathological" phenomena that arise in general measure theory.

Measures in topological spaces.[edit]

The study of measures in topological spaces is usually concerned with measures defined on sets connected in some way or another with the topology of the underlying space. One of the typical approaches is the following. Let $X$ be an arbitrary topological space and let $\mathcal{Z}$ be the class of subsets of the form $f^{-1}(F)$, where $f$ is a continuous real-valued function on $X$ and $F\subset {\mathbf{R}}^1$ is a closed set. Let $\mathfrak{A}$ be the algebra generated by the class $\mathcal{Z}$ and let $\mathcal{B}$ be the $\sigma$- algebra generated by $\mathcal{Z}$( $\mathcal{B}$ is called the $\sigma$- algebra of Baire sets, cf. also Algebra of sets). Now let $\mathcal{M}$ be the class of totally-finite finitely-additive measures $m$ on $\mathfrak{A}$ that are regular in the sense that

$$m(E)=sup\{m(Z):Z\subset E,Z\in \mathcal{Z}\}$$

for all $E\in \mathfrak{A}$. In $\mathcal{M}$ one distinguishes the subclasses ${\mathcal{M}}_{\sigma }$, ${\mathcal{M}}_{\tau }$ and ${\mathcal{M}}_t$ formed by the (finitely-additive) measures possessing additional smoothness properties. By definition, $\mu \in {\mathcal{M}}_{\sigma }$ if $\mu (Z_n)\downarrow 0$ for every sequence $Z_n\downarrow \mathrm{\varnothing }$, $Z_n\in \mathcal{Z}$( this property is equivalent to the countable additivity of $\mu$; the measures from ${\mathcal{M}}_{\sigma }$ admit unique extensions to $\mathcal{B}$ and hereafter it is assumed that they are given on $\mathcal{B}$); $\mu \in {\mathcal{M}}_{\tau }$ if $\mu ({\mathcal{Z}}_{\alpha })\downarrow 0$ for every net $Z_{\alpha }\downarrow \mathrm{\varnothing }$, $Z_{\alpha }\in \mathcal{Z}$; and $\mu \in {\mathcal{M}}_t$ if for every $ϵ>0$ there is a compact set $K$ such that $\mu (E)<ϵ$ whenever $E\subset X\setminus K$, $E\in \mathfrak{A}$.

The inclusions $\mathcal{M}\supset {\mathcal{M}}_{\sigma }\supset {\mathcal{M}}_{\tau }\supset {\mathcal{M}}_t$ hold. The elements of ${\mathcal{M}}_{\sigma }$ are called Baire measures.

There is an intimate connection between the measures belonging to $\mathcal{M}$ and the linear functionals on the space $C(X)$ of bounded continuous functions on $X$. Namely, the formula

$$\mathrm{\Lambda }(f)=\underset{X}{\int }fdm$$

establishes a one-to-one correspondence between the finitely-additive measures $m\in \mathcal{M}$ and the non-negative linear functionals $\mathrm{\Lambda }$ on $C(X)$( non-negative means that $\mathrm{\Lambda }(f)\ge 0$ whenever $f(x)\ge 0$, $x\in X$). Moreover, for every set $Z\in \mathcal{Z}$,

$$m(Z)=\inf \{\mathrm{\Lambda }(f):{\chi }_Z\le f\le 1\},$$

where ${\chi }_Z$ is the indicator function of $Z$. This correspondence takes the measures from ${\mathcal{M}}_{\sigma }$ into $\sigma$- smooth functionals $\mathrm{\Lambda }$( i.e. functionals $\mathrm{\Lambda }$ with the property that $\mathrm{\Lambda }(f_n)\to 0$ if $f_n\downarrow 0$ in $C(X)$), the measures from ${\mathcal{M}}_{\tau }$ into $\tau$- smooth functionals $\mathrm{\Lambda }$( i.e. functionals such that $\mathrm{\Lambda }(f_{\alpha })\to 0$ for every net $f_{\alpha }\downarrow 0$ in $C(X)$), and the measures from ${\mathcal{M}}_t$ into dense functionals $\mathrm{\Lambda }$( i.e. with the property that $\mathrm{\Lambda }(f_{\alpha })\to 0$ for every net $f_{\alpha }$ in $C(X)$ such that $\Vert f_{\alpha }\Vert \le 1$ for all $\alpha$ and $f_{\alpha }\to 0$ uniformly on compact subsets; here $\Vert \cdot \Vert$ is the uniform norm).

The space $\mathcal{M}$ is usually endowed with the weak topology $w$, in which a basis of neighbourhoods consists of the sets of the form

$$U(m_0;f_1\dots f_n,ϵ)=$$

$$=\{m:|\underset{X}{\int }{f}_k(dm-d{m}_0)|<ϵ,k=1\dots n,f_1\dots f_n\in C(X)\}.$$

With the topology $w$, $\mathcal{M}$ is a completely-regular Hausdorff space. Convergence in the topology $w$ is usually denoted by the symbol $\Rightarrow$. For the convergence of a net $m_{\alpha }$ to $m$: $m_{\alpha }\Rightarrow m$, it is necessary and sufficient that $m_{\alpha }(X)\to m(X)$ and $limsup{m}_{\alpha }(Z)\le m(Z)$ for all $Z\in \mathcal{Z}$. Another necessary and sufficient condition for the convergence $m_{\alpha }\Rightarrow m$ is that $m_{\alpha }(E)\to m(E)$ for all $E\in \mathfrak{A}$ such that there are $Z_1,Z_2\in \mathcal{Z}$ with $X\setminus E\subset Z_1$, $E\subset Z_2$, and $m(Z_1\cap Z_2)=0$. If the space $X$ is completely regular and Hausdorff, then ${\mathcal{M}}_{\tau }$ is metrizable if and only if $X$ is metrizable. If $X$ is metrizable, then ${\mathcal{M}}_{\tau }$ admits a metric in which it is separable if and only if $X$ is separable, and it admits a metric in which it is complete if and only if $X$ has a complete metric. If $X$ is metrizable, then ${\mathcal{M}}_{\sigma }$ is metrizable if and only if it is metrizable by the Lévy–Prokhorov metric.

The space ${\mathcal{M}}_{\sigma }$ is sequentially closed in $\mathcal{M}$( Aleksandrov's theorem). A set $A\subset \mathcal{M}$ is called tight if $sup\{m(X):m\in A\}<\mathrm{\infty }$ and if for every $ϵ>0$ there is a compact set $K$ such that $m(E)<ϵ$ for all $E\subset X\setminus K$, $m\in A$ and $E\in \mathfrak{A}$. If $A\subset {\mathcal{M}}_{\sigma }$ is tight, then $A$ is relatively compact in ${\mathcal{M}}_{\sigma }$; conversely, if $X$ is metrizable and topologically complete, then $A\subset {\mathcal{M}}_{\sigma }$ is relatively compact, and if every measure in $A$ is concentrated on some separable subset of $X$, then $A$ is tight (Prokhorov's theorem).

Under certain conditions the elements of ${\mathcal{M}}_{\sigma }$ can be extended to Borel measures, i.e. measures defined on the $\sigma$- algebra of Borel sets (see Borel set; Borel measure). For example, if $X$ is a normal countably-paracompact Hausdorff space, then every measure $\mu \in {\mathcal{M}}_{\sigma }$ admits a unique extension to a regular Borel measure. If $X$ is completely regular and Hausdorff, then every $\tau$- smooth (tight) Baire measure admits a unique extension to a $\tau$- smooth (tight) Borel measure.

The support of a Baire (Borel) measure is the smallest set $Z\in \mathcal{Z}$( respectively, the smallest closed set) the measure of which is equal to the measure of the whole space. Every $\tau$- smooth measure has a support.

Often, when measures in topological spaces (especially in locally compact Hausdorff spaces) are considered, it is assumed that the Borel and Baire measures are given on less-wide classes of sets, more precisely — on $\sigma$- rings generated by compact sets and, respectively, compact $G_{\delta }$- sets.

Let $G$ be a locally compact Hausdorff topological group. A left Haar measure on $G$ is a measure defined on the $\sigma$- ring generated by all compact subsets that does not vanish identically and is such that $\mu (xE)=\mu (E)$ for all $x\in G$ and $E$ in the domain of $\mu$. A right Haar measure is defined in the same manner but with the condition $\mu (xE)=\mu (E)$ replaced by $\mu (Ex)=\mu (E)$. On any group of the type considered a left Haar measure exists and is unique (up to a multiplicative positive constant). Every left Haar measure is regular in the sense that $\mu (E)=sup\{\mu (K):K\subset E\}$, where $K$ are compact sets. The right Haar measure has analogous properties. The Lebesgue measure on ${\mathbf{R}}^k$ is a particular case of the Haar measure. See also Measure in a topological vector space.

Isomorphism of measure spaces.[edit]

Let $(X,\mathcal{S},\mu )$ be a measure space. Call two sets $E,E^{\prime }\in \mathcal{S}$ $\mu$- equal (written $E=E^{\prime }$ $[\mu ]$) if $\mu (E\mathrm{\Delta }{E}^{\prime })=0$( where $E\mathrm{\Delta }{E}^{\prime }$ denotes the symmetric difference of $E$ and $E^{\prime }$, cf. Symmetric difference of sets). Denote by ${\mathcal{S}}_{\mu }$ the class of sets $\mathcal{S}$ with this equality relation. In ${\mathcal{S}}_{\mu }$ the set-theoretic operations, performed a finite (or countable) number of times are correctly defined: for example, if $E_1=E_1^{\prime }$ $[\mu ]$ and $E_2=E_2^{\prime }$ $[\mu ]$, then $E_1\cup E_2=E_1^{\prime }\cup E_2^{\prime }$ $[\mu ]$. The measure $\mu$ is carried over, in an obvious manner, to ${\mathcal{S}}_{\mu }$.

Let ${\tilde{\mathcal{S}}}_{\mu }$ be the subset of ${\mathcal{S}}_{\mu }$ consisting of the sets of finite measure. The function $\rho (E,E^{\prime })=\mu (E\mathrm{\Delta }{E}^{\prime })$ on ${\tilde{\mathcal{S}}}_{\mu }\times {\tilde{\mathcal{S}}}_{\mu }$ is a metric. The measure space $(X,\mathcal{S},\mu )$ is said to be separable if the space ${\tilde{\mathcal{S}}}_{\mu }$ with metric $\rho$ is separable. If $(X,\mathcal{S},\mu )$ is a space with a $\sigma$- finite measure and the $\sigma$- ring $\mathcal{S}$ is countably generated (i.e. there is a countable family $\{E_n\}\subset \mathcal{S}$ such that $\mathcal{S}$ is the smallest $\sigma$- ring that contains this family), then the metric space ${\tilde{\mathcal{S}}}_{\mu }$ is separable.

Two measure spaces, $(X_1,{\mathcal{S}}_1,{\mu }_1)$ and $(X_2,{\mathcal{S}}_2,{\mu }_2)$ are said to be isomorphic if there is a one-to-one mapping $\varphi$ of $({\mathcal{S}}_1{)}_{{\mu }_1}$ onto $({\mathcal{S}}_2{)}_{{\mu }_2}$ such that

$$\varphi (E\setminus F)=\varphi (E)\setminus \varphi (F),\varphi (E\cup F)=\varphi (E)\cup \varphi (F)$$

and

$${\mu }_1(E)={\mu }_2(\varphi (E))\text{ for all }E,F\in ({\mathcal{S}}_1{)}_{{\mu }_1}.$$

Now, let $(X,\mathcal{S},\mu )$ be an arbitrary space with a totally-finite measure. There is a partition of $X$ into disjoint sets $X_n\in \mathcal{S}$, $n=1,2\dots$ such that the restriction of $\mu$ to $X_n$ is isomorphic either to a measure concentrated at one point or to a measure which is equal, up to a positive factor, to the direct product $\prod _{i\in I}(U_i,{\mathcal{U}}_i,u_i)$, where $U_i=\{0,1\}$, $u_i(\{0\})=u_i(\{1\})=1/2$, and the set $I$ may have arbitrary cardinality (the Maharan–Kolmogorov theorem). If $(X,\mathcal{S},\mu )$ is separable, non-atomic and $\mu (X)=1$, then it is isomorphic to the space $\prod _{i\in I}(U_i,{\mathcal{U}}_i,u_i)$ with $I$ countable, which in turn is isomorphic to the unit interval with the Lebesgue measure.

Side by side with the theory of measures regarded as functions on subsets of some set, the theory of measures as functions on the elements of a Boolean ring (or on a Boolean algebra) has been developed; these theories are in many respects parallel. Another widespread construction of measures goes back to W. Young and P. Daniell (see [12]). Theories dealing with measures with real or complex values, or with values belonging to some algebraic structure, were developed in addition to the theory of positive measures.

References[edit]

Comments[edit]

Properties 1) and 2) listed under the heading "Properties of measure spaces" are usually called Fatou's lemma, cf. Fatou theorem.

The procedure for extending a measure, as described under the heading "Extension of measures" , is due to C. Carathéodory, and one often speaks of Carathéodory extension, with the accompanying phrases Carathéodory extension theorem and Carathéodory outer (inner) measure (cf. Carathéodory measure).

Recall that a ring (respectively, a $\sigma$- ring) $\mathcal{A}$ of subsets of a set $X$ such that $A\in \mathcal{A}$ implies $X\setminus A\in \mathcal{A}$, is called a Boolean algebra or an algebra (respectively, a $\sigma$- algebra or a $\sigma$- field, cf. also Algebra of sets). Usually, in a measure space $(X,\mathcal{S},\mu )$ the $\sigma$- ring $\mathcal{S}$ can be proved to be a $\sigma$- field (this holds, in particular, if $\mu (X)<\mathrm{\infty }$).

The phrase "totally (s-) finite" is seldom used.

Borel has given very nice ideas in order to construct the measure ${\lambda }^{\prime }$, but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of $\bar{\lambda }$.

A product space is also often written as a (kind of) tensor product: $(X_1\times X_2,{\mathcal{S}}_1\otimes {\mathcal{S}}_2,{\mu }_1\otimes {\mu }_2)$.

A family of measurable spaces $(X_i,{\mathcal{S}}_i{)}_i$ with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on $\prod X_i$, if it exists, is called the projective limit; it exists if $I$ is countable (the Ionescu–Tulcea theorem, cf. [5]).

Suppose that $X$ is a topological space and $\mathcal{S}$ is its Borel $\sigma$- field; then $(X,\mathcal{S},\mu )$ is perfect for every finite measure $\mu$ if $X$ is a Polish space or, more generally, a Luzin space (in which case $(X,\mathcal{S})$ is often called a standard measurable space) or, still more generally, a Suslin space (in which case $(X,\mathcal{S})$ is sometimes called a Blackwell measurable space) (cf. (the editorial comments to) Descriptive set theory).

The converse part of Prokhorov's theorem is not true when $X$ is the space of rational numbers, or, more generally, when $X$ is a Luzin space which is not Polish. See [a1].

In the abstract setting, whenever $({\mu }_n)$ is a sequence of finite measures on $(X,\mathcal{S})$, where $\mathcal{S}$ is a $\sigma$- field, such that

$$m(A)=\underset{n}{lim}{\mu }_n(A)$$