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An integral with respect to a [[Radon measure]]. [[Category:Measure and integration]]

...nected with a measure defined on an abstract set are the sets on which the measure under discussion is defined. ...p"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated f

An integral of a vector-valued function with respect to a scalar measure, which is a so-called weak integral. It was introduced by B.J. Pettis [[#Re ...ubset E$ if for any $f\in X^*$ the function $f[x(t)]$ is integrable on $M$ and if there exists an element $x(M)\in X$ such that

[[Category:Measure and integration]]

Please remove this comment and the {{TEX|auto}} line below, ...For example, instead of a measure, a [[Pre-measure|pre-measure]] (or quasi-measure) can be used, that is, an additive set function defined on the algebra of a

[[Category:Classical measure theory]] The term might refer to different objects in classical measure theory.

$#C+1 = 75 : ~/encyclopedia/old_files/data/I052/I.0502240 Invariant integration Please remove this comment and the {{TEX|auto}} line below,

$#C+1 = 21 : ~/encyclopedia/old_files/data/F038/F.0308480 Feynman measure Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ''measurable partition, of a [[Measure space|measure space]] $ ( M, \mu ) $''

$#C+1 = 49 : ~/encyclopedia/old_files/data/Q076/Q.0706560 Quasi\AAhinvariant measure Please remove this comment and the {{TEX|auto}} line below,

[[Category:Set functions and measures on topological spaces]] ...mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with [[Borel measure]]). The support of $\mu$ (usually denoted by ${\rm supp}\, (\mu)$) is the c

[[Category:Classical measure theory]] ''inner regular measure, Radon measure''

Please remove this comment and the {{TEX|auto}} line below, A finitely-additive [[Measure|measure]] with real or complex values on some space $ \Omega $

...is a closed subset $F\subset X$ such that $\mu(X\setminus F)<\varepsilon$ and $f$ is continuous on $F$ (Luzin's theorem). ..."top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)</TD></

...\bigcup_i U_i$ and $\sum_i \mu(U_i) < \epsilon$, where $\mu$ is [[Lebesgue measure]]. ...additive function]] on $\mathcal{E}$. Such a function is called a content, and $\gamma(A)$ is the content of $A$.

Please remove this comment and the {{TEX|auto}} line below, and $ u ( t, \omega ) $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ..._ { n } ( E ) | < \varepsilon$ whenever $E \in \Sigma$, $n \in \mathbf N$ and $\lambda ( E ) < \delta$.

...izes to multidimensional integrals the usual [[Integration by substitution|integration by substitution]] of integrals in one variable. Let $U$ and $V$ be open sets in $\mathbb R^n$, $\Phi: U \to V$ be a [[Diffeomorphism|di

[[Category:Classical measure theory]] ...on of a '''nonnegative measure''' we refer to [[Jordan decomposition (of a measure)]].

...psilon>0$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set ...and only if it becomes continuous if one neglects a set of arbitrary small measure.

Please remove this comment and the {{TEX|auto}} line below, An integral in which there is a successive integration with respect to different variables, i.e. an integral of the form

[[Category:Classical measure theory]] A concept of classical measure theory.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Bayesi

[[Category:Classical measure theory]] ...ebra]] $\mathcal{B}$ of subsets of a set $X$ and a [[Signed measure|signed measure]]

Please remove this comment and the {{TEX|auto}} line below, ...valign="top">[a1]</TD> <TD valign="top"> M.M. Rao, "Measure theory and integration" , Wiley (Interscience) (1987) pp. Chapt. 7</TD></TR></table>

[[Category:Classical measure theory]] ...bra of sets|$\sigma$-algebra]] $\mathcal{B}$ of subsets of $X$, then $\mu$ and $\nu$ are said to be singular

Please remove this comment and the {{TEX|auto}} line below, be such a group and let $ \alpha $

...all compact subsets, taking finite values on all compact subsets of $ G $, and satisfying either the condition of '''left-invariance''': ...rdingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is $ \mu $-'''regular''', i.e.,

Please remove this comment and the {{TEX|auto}} line below, ...and most important case of a product of spaces, see the article [[Measure|Measure]]. A more general construction is given below. Let $ I $

Please remove this comment and the {{TEX|auto}} line below, ...iple Riemann integral is based on the concept of a [[Jordan measure|Jordan measure]] $ \mu $.

...Riemannian manifolds]] and which introduces a natural concept of [[Measure|measure]] on the manifold. ...basis) of $w_j$. Moreover, $|\omega (v_1, \ldots, v_n)|$ is the [[Lebesgue measure]] (i.e. the $n$-dimensional volume) of the [[Parallelopipedon|parallelopipe

[[Category:Set functions and measures on topological spaces]] The terminology ''Borel measure'' is used by different authors with different meanings:

[[Category:Classical measure theory]] A central theorem in classical measure theory, sometimes called

''inner regular measure'' [[Category:Classical measure theory]]

[[Category:Classical measure theory]] ===Definition and properties===

[[Category:Classical measure theory]] ...em on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. In literature it is somet

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...B.J. Pettis needed to use this result but noted that no proof was supplied and then proceeded to give a proof ([[#References|[a6]]]; the proof is very sim

[[Category:Classical measure theory]] A concept in measure theory (see also [[Absolute continuity]]). If $\mu$ and $\nu$ are two measures on a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ o

[[Category:Classical measure theory]] A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following statement.

[[Category:Classical measure theory]] An outer measure is a [[Set function|set function]] $\mu$ such that

Please remove this comment and the {{TEX|auto}} line below, ...eferences|[a10]]], [[#References|[a11]]], the survey [[#References|[a2]]], and [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]).

...area bounded by the graph of a function $f$ on an interval $a\leq x\leq b$ and the $x$-axis (the problem of calculating the work performed by a force over ...ral, the indefinite and the definite integral. The study of the properties and calculation of these interrelated forms of the integral constitutes the pro

$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space Please remove this comment and the {{TEX|auto}} line below,

''generalized measure'', ''real valued measure'', ''charge'' [[Category:Classical measure theory]]

[[Category:Classical measure theory]] A concept of classical measure theory.

[[Category:Classical measure theory]] ...ed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ame{supp} \phi_{j} \subset K$ for all $j$ and the coefficients of $\phi_j$ and all their derivatives tend uniformly to those of $\phi$.

Please remove this comment and the {{TEX|auto}} line below, ...perties of functions are studied on the basis of the idea of the [[Measure|measure]] of a set.

Please remove this comment and the {{TEX|auto}} line below, ...) and especially Archimedes (3rd century B.C.). Its subsequent application and perfection is associated with the names of several scholars of the 15th–1

Please remove this comment and the {{TEX|auto}} line below, ...homogeneous|Random field, homogeneous]]). Stationary stochastic processes and homogeneous random fields for which the Fourier transform of the covariance

Please remove this comment and the {{TEX|auto}} line below, be a self-adjoint and $ U $

[[Category:Classical measure theory]] .... Let $(X,\mu)$ be a space with a non-negative complete countably-additive measure $\mu$ (cf.

Please remove this comment and the {{TEX|auto}} line below, and $ b _ {i} $

Please remove this comment and the {{TEX|auto}} line below, ...n integral]] and defined another integral that is equivalent to the Denjoy and Perron integrals. This integral is known as the gauge integral, the Riemann

[[Category:Classical measure theory]] The term Hausdorff measures is used for a class of [[Outer measure|outer measures]] (introduced for the first time by Hausdorff in {{Cite|Ha}}

\mathbb R$ and a continuously differentiable function $v:[a,b]\to \mathbb R$. If $U$ is a primitive of $u$, the integration by parts formula for the definite integral $\int_a^b

Please remove this comment and the {{TEX|auto}} line below, where the integration is performed over the straight line $ \Pi _ \alpha $

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...mputation. Continuous computational complexity studies continuous problems and tends to use the real number model; see [[#References|[a1]]], [[#References

...rals are equal, their common value is the Burkill integral of $F$ over $R$ and is denoted by $\int_RF$. If $F$ is integrable over $R$, then $F$ is integra ...e class must permit a subdivision into sets also of this class that have a measure as small as one pleases. The Burkill integral can then be defined for any s

[[Category:Classical measure theory]] ...roperty of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional

...1]</TD> <TD valign="top"> H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston (1972) pp. 332 (Translated from G ...n="top">[a5]</TD> <TD valign="top"> M.M. Rao, "Stochastic processes and integration" , Sijthoff &amp; Noordhoff (1979)</TD></TR>

[[Category:Classical measure theory]] ...asurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has [[Density of a set|Lebesgue density]] $1$.

Please remove this comment and the {{TEX|auto}} line below, has Jordan measure zero, and which are Riemann integrable on any segment not containing points of $ E

...antity $K$ equal to the product of the principal (normal) curvatures $k_1$ and $k_2$ calculated at the point on the surface: $K=k_1k_2$; it is usually cal ...nterior curvature (the sectional curvature of the two-dimensional surface) and the exterior curvature (the sectional curvature of the ambient space in the

Let $U\subset \mathbb R^n$, $f: U\to \mathbb R^m$ and assume that $f$ is differentiable at the point $y\in U$. where $(f^1, \ldots, f^m)$ are the coordinate functions of $f$ and $x_1,\ldots, x_n$ denote the standard system of coordinates in

[[Category:Classical measure theory]] ...(X,\A,\mu)$ is often shortened to $(X,\mu)$ and one says that $\mu$ is a measure on $X$; sometimes the notation is shortened to $X$.

Please remove this comment and the {{TEX|auto}} line below, and 2) $ u( z ^ {0} + \lambda a) $

[[Category:Classical measure theory]] ...that is, a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes]] probability measure);

[[Category:Classical measure theory]] ...between the [[Measurable space|measurable spaces]] $(X, \mathcal{B} (X))$ and $(Y, \mathcal{B} (Y))$, see also [[Measurable mapping]].

[[Category:Classical measure theory]] ...easurable set $E\subset \mathbb R$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has [[Density of a set|Lebesgue density]] $1$

...uestions of representation and approximation of functions, and their local and global properties. The modern theory of functions of a real variable typica ...y, N.I. Lobachevskii, P. Dirichlet, B. Riemann, P.L. Chebyshev, C. Jordan, and others).

...kernel $\ln 1/|x-y|$, where $|x-y|$ is the distance between the points $x$ and $y$ of the Euclidean plane $\mathbf R^2$, that is, a potential of the form ...lue problems (see [[Exterior and interior boundary value problems|Exterior and interior boundary value problems]]). The main applications of the logarithm

[[Category:Classical measure theory]] ...le subset of the euclidean space coincides with a Borel set up to a set of measure zero. More precisely (cp. with Proposition 15 of Chapter 3 in {{Cite|Ro}}):

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...al distribution if $X := \Gamma X$ for all $\Gamma \in \mathcal{O} ( p )$; and

Please remove this comment and the {{TEX|auto}} line below, ...problems in]]) to the phase space (space of states) of any physical system and a method of averaging physical (observable) quantities related to this syst

[[Category:Classical measure theory]] ...ure|signed measures]]. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as:

...ariski topology), is not a topological group with respect to this topology and therefore is not a compact group. ...ry norm of the fields $ \mathbf C $ and $ \mathbf R $ , respectively) and, more generally, any compact real Lie group.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. finite measure $ \mu $

[[Category:Classical measure theory]] ...\in E$ has [[Density of a set|density]] $1$ with respect to the [[Lebesgue measure]] $\lambda$, that is

...$m$ be the Wiener measure, which is the [[Probability measure|probability measure]] on $(C,\mathcal B)$ making the coordinate mappings a [[Brownian motion|Br a) they are met in a great number of applications, including filtering and mathematical finance;

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...a$-algebra of $\mathcal{S} ^ { \prime } ( \mathbf{R} )$ and $d \mu$ is the measure determined by

Please remove this comment and the {{TEX|auto}} line below, ...integral equation|Singular integral equation]]), in the study of interior and [[Boundary properties of analytic functions|boundary properties of analytic

''measure of a set'' ...in the theory of functions of a real variable in connection with the study and improvement of the notion of an [[Integral|integral]].

Please remove this comment and the {{TEX|auto}} line below, ...s) is well established in the literature since the activity of V.P. Maslov and his collaborators (see e.g. [[#References|[a1]]], [[#References|[a2]]], [[#

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...or is (weakly) compact if it takes bounded sets into (weakly) compact sets and completely continuous if it takes weakly compact sets into norm-compact set

...rnel $1/|x-y|^{N-2}$, where $|x-y|$ is the distance between two points $x$ and $y$ of the Euclidean space $\mathbf R^N$, $N\geq3$, that is, an integral of ...|Radon measure]] $\mu$ on $\mathbf R^N$ with compact support $S$. When the measure $\mu$ is non-negative, the Newton potential \eqref{1} is a superharmonic fu

Please remove this comment and the {{TEX|auto}} line below, such that: 1) the co-mass (cf. [[Mass and co-mass|Mass and co-mass]]) $ | \omega | _ {0} \leq N _ {1} $

Please remove this comment and the {{TEX|auto}} line below, ...tions of analytic sheaves, analytic sheaves, analytic subsets, holomorphic and meromorphic mappings, from the complement $ X \setminus A $

Please remove this comment and the {{TEX|auto}} line below, and with centre at the coordinate origin has the form

Please remove this comment and the {{TEX|auto}} line below, with respect to a measure $ \mu ( d \omega ) $

Please remove this comment and the {{TEX|auto}} line below, In this case the flow has an [[Invariant measure|invariant measure]] $ \mu ( A) = \int _ {A} \phi $,

...=x_0<x_1<\dots<x_n=b$ is a partition (subdivision) of the interval $[a,b]$ and $\Delta x_i = x_i-x_{i-1}$, where $i=1,\dots,n$. The sum .... The limit is known as the definite Riemann integral of $f$ over $[a,b]$, and is written as

Please remove this comment and the {{TEX|auto}} line below, is also left (right) almost-periodic, and the family $ f ( a x b ) $,

Please remove this comment and the {{TEX|auto}} line below, is a locally convex space if and only if the topology of $ E $

Please remove this comment and the {{TEX|auto}} line below, ...n [[Orthonormal system|orthonormal system]] of functions with respect to a measure $ \mu $:

Please remove this comment and the {{TEX|auto}} line below, ...y has been constructed for various spaces, primarily Euclidean, projective and homogeneous spaces.

Please remove this comment and the {{TEX|auto}} line below, is an orthogonal random measure on $ \Lambda $(

...series, linear operators, random processes, and in the theory of extremal and approximation problems. ...www.encyclopediaofmath.org/legacyimages/h/h046/h046320/h04632022.png" />), and

Please remove this comment and the {{TEX|auto}} line below, with equality if and only if $ K $

Please remove this comment and the {{TEX|auto}} line below, 1) the vector space structure and the partial order are compatible. i.e. from $ x, y, z \in X $

Please remove this comment and the {{TEX|auto}} line below, on which two structures are given — a group structure and a topology, such that the group operations are continuous. Specifically, th

Please remove this comment and the {{TEX|auto}} line below, ...es of decision-making, used in [[Operations research|operations research]] and game theory (cf. [[Games, theory of|Games, theory of]]). It is implicit in

Please remove this comment and the {{TEX|auto}} line below, ...stic assigned to planar figures of a certain class (for example, polygons) and having the following properties: 1) the area is non-negative; 2) the area i

...t of convergence arises, for example, in the study of mathematical objects and their approximation by simpler objects. Thus, in order to calculate the are ...e to calculate this solution with the required accuracy. Both for ordinary and partial differential equations there are various convergent difference meth

Please remove this comment and the {{TEX|auto}} line below, ...which boundary properties of functions are studied in terms of topological and metric properties of various cluster sets.

...iven series) of some [[Linear topological space|linear topological space]] and a certain infinite set of their partial sums (called the partial sums of th and $ \{ s _{n} \} $

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...$ and let $d \sigma _ { r }$ be the normalized [[Lebesgue measure|Lebesgue measure]] on $S ( x , r )$. One version of the classical converse of Gauss' mean-va

Please remove this comment and the {{TEX|auto}} line below, ...white noise analysis has developed into a viable framework for stochastic and infinite-dimensional analysis [[#References|[a4]]]–[[#References|[a6]]],

Please remove this comment and the {{TEX|auto}} line below, and the vector $ k _ {M} $.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...on of a non-linear differential equation with a Gaussian white noise input and an observation process whose differential is a non-linear function of the s

Please remove this comment and the {{TEX|auto}} line below, unknowns $ p = ( p _ {1} \dots p _ {n} ) $ ("generalized momenta") and $ q = ( q _ {1} \dots q _ {n} ) $ ("generalized coordinates"), of the for

Please remove this comment and the {{TEX|auto}} line below, and let $ p $

Please remove this comment and the {{TEX|auto}} line below, and the independent observations $ \phi $

Please remove this comment and the {{TEX|auto}} line below, ...eories, in string theory, where both kinds of transformations are present, and in two-dimensional conformal field theories.

...e quotient space $\Gamma\setminus G$ has finite volume in the sense of the measure induced by the left-invariant [[Haar measure|Haar measure]] on the group $G$. The concept of lattices includes that of uniform discre

Please remove this comment and the {{TEX|auto}} line below, with respect to the Lebesgue measure $ dV $.

...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic in the area of probability and statistics.

Please remove this comment and the {{TEX|auto}} line below, where the integral is taken with respect to the Haar measure.

Please remove this comment and the {{TEX|auto}} line below, ...onnection with work on the construction of atomic reactors, J. von Neumann and S. Ulam suggested using the apparatus of probability theory in the computer

...ucted analogously. A further generalization leads to Fourier coefficients and Fourier series for elements of a Hilbert space. ...r series in the trigonometric system has been most thoroughly developed, and these were the first examples of Fourier series. If one has in mind Fourie

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ot simultaneously be faithful extensions of the distributional derivatives and the pointwise product of continuous functions. Thus, a multiplication of di

Please remove this comment and the {{TEX|auto}} line below, and let $ a = x _ {0} < \dots < x _ {i - 1 } < x _ {i} < \dots < x _ {n} = b

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...not map the subspace corresponding to the continuous spectrum into itself, and, as a consequence, the model becomes solvable in the sense described below.

Please remove this comment and the {{TEX|auto}} line below, and $ f \in L _ {2} $;

Please remove this comment and the {{TEX|auto}} line below, ...oy analytical tools in this theory (in particular, infinitesimal methods), and also to considerably enlarge the class of "natural" group algebras (functio

Please remove this comment and the {{TEX|auto}} line below, ...theory of elementary functions, the theory of series, Riemann and Lebesgue integration, the theory of functions of a complex variable, the theory of generalized f

Please remove this comment and the {{TEX|auto}} line below, ...line or (hyper)plane integrals. Important issues are existence, uniqueness and stability of inversion procedures, as well as the development of efficient

...ariable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$. ...ng all numbers and choosing one of them; or finally a dependence can exist and remain unknown" . The definition of a function as a correspondence between

...://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l .../www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934011.png" />, and (in the case <img align="absmiddle" border="0" src="https://www.encyclopedi

...yen, M.V. Karasev, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, and V.P. Maslov [[#References|[a32]]], [[#References|[a33]]], [[#References|[a6 ...ent mathematical studies are in some sense related to such a point of view and have given new interesting prospects in pure mathematics. See e.g. [[#Refer

Please remove this comment and the {{TEX|auto}} line below, and therefore surfaces with $ H = 0 $

Please remove this comment and the {{TEX|auto}} line below, be a linear differential operator, and let

Please remove this comment and the {{TEX|auto}} line below, ...pment of mathematics. The practical activities of mankind on the one hand, and the internal demands of mathematics on the other, have determined the devel

Please remove this comment and the {{TEX|auto}} line below, ...ces|[2]]]; engineers had discovered the idea of a feedback law (see below) and found means for its realization.