An integral with respect to a [[Radon measure]].
[[Category:Measure and integration]]
100 bytes (14 words) - 21:22, 15 November 2014
...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
2 KB (289 words) - 20:54, 25 October 2014
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
2 KB (384 words) - 09:44, 22 August 2014
[[Category:Measure and integration]]
346 bytes (45 words) - 12:28, 9 November 2014
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
6 KB (847 words) - 22:12, 5 June 2020
[[Category:Classical measure theory]]
The term might refer to different objects in classical measure theory.
4 KB (584 words) - 09:27, 7 December 2012
$#C+1 = 75 : ~/encyclopedia/old_files/data/I052/I.0502240 Invariant integration
Please remove this comment and the {{TEX|auto}} line below,
5 KB (743 words) - 22:13, 5 June 2020
$#C+1 = 21 : ~/encyclopedia/old_files/data/F038/F.0308480 Feynman measure
Please remove this comment and the {{TEX|auto}} line below,
2 KB (349 words) - 19:39, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
''measurable partition, of a [[Measure space|measure space]] $ ( M, \mu ) $''
5 KB (757 words) - 08:00, 6 June 2020
$#C+1 = 49 : ~/encyclopedia/old_files/data/Q076/Q.0706560 Quasi\AAhinvariant measure
Please remove this comment and the {{TEX|auto}} line below,
4 KB (564 words) - 17:20, 11 March 2021
[[Category:Set functions and measures on topological spaces]]
...mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with [[Borel measure]]). The support of $\mu$ (usually denoted by ${\rm supp}\, (\mu)$) is the c
3 KB (393 words) - 16:15, 29 November 2012
[[Category:Classical measure theory]]
''inner regular measure, Radon measure''
4 KB (498 words) - 22:40, 31 December 2017
Please remove this comment and the {{TEX|auto}} line below,
A finitely-additive [[Measure|measure]] with real or complex values on some space $ \Omega $
3 KB (492 words) - 04:40, 9 May 2022
...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></
2 KB (395 words) - 04:33, 23 July 2015
...\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 $A$.
2 KB (378 words) - 15:57, 8 October 2017
Please remove this comment and the {{TEX|auto}} line below,
and $ u ( t, \omega ) $
8 KB (1,147 words) - 13:18, 26 March 2023
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 $\lambda ( E ) < \delta$.
4 KB (614 words) - 07:39, 24 January 2024
...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
3 KB (384 words) - 14:28, 14 December 2012
[[Category:Classical measure theory]]
...on of a '''nonnegative measure''' we refer to [[Jordan decomposition (of a measure)]].
3 KB (419 words) - 12:56, 2 December 2012
...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.
3 KB (436 words) - 19:43, 28 December 2017
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
7 KB (1,043 words) - 01:21, 7 May 2022
[[Category:Classical measure theory]]
A concept of classical measure theory.
3 KB (373 words) - 17:15, 18 August 2012
...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
6 KB (908 words) - 18:44, 21 March 2024
[[Category:Classical measure theory]]
...ebra]] $\mathcal{B}$ of subsets of a set $X$ and a [[Signed measure|signed measure]]
3 KB (403 words) - 10:49, 10 December 2012
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>
977 bytes (144 words) - 08:05, 6 June 2020
[[Category:Classical measure theory]]
...bra of sets|$\sigma$-algebra]] $\mathcal{B}$ of subsets of $X$, then $\mu$ and $\nu$ are said to be singular
3 KB (399 words) - 17:27, 18 August 2012
Please remove this comment and the {{TEX|auto}} line below,
be such a group and let $ \alpha $
3 KB (413 words) - 08:01, 6 June 2020
...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.,
6 KB (971 words) - 20:18, 23 April 2017
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 $
4 KB (573 words) - 17:45, 4 June 2020
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 $.
11 KB (1,648 words) - 16:53, 20 January 2024
...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
7 KB (1,061 words) - 06:58, 29 June 2014
[[Category:Set functions and measures on topological spaces]]
The terminology ''Borel measure'' is used by different authors with different meanings:
5 KB (764 words) - 09:39, 16 August 2013
[[Category:Classical measure theory]]
A central theorem in classical measure theory, sometimes called
3 KB (451 words) - 17:37, 18 August 2012
''inner regular measure''
[[Category:Classical measure theory]]
4 KB (638 words) - 19:35, 1 January 2021
[[Category:Classical measure theory]]
===Definition and properties===
5 KB (691 words) - 09:52, 6 July 2013
[[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
4 KB (595 words) - 17:27, 18 October 2012
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
5 KB (714 words) - 15:30, 1 July 2020
[[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
6 KB (957 words) - 07:56, 15 December 2016
[[Category:Classical measure theory]]
A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following statement.
3 KB (484 words) - 12:09, 14 December 2012
[[Category:Classical measure theory]]
An outer measure is a [[Set function|set function]] $\mu$ such that
9 KB (1,397 words) - 10:08, 16 August 2013
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]]]).
9 KB (1,288 words) - 08:49, 26 March 2023
...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
14 KB (2,170 words) - 17:21, 14 February 2020
$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
Please remove this comment and the {{TEX|auto}} line below,
7 KB (1,017 words) - 08:00, 6 June 2020
''generalized measure'', ''real valued measure'', ''charge''
[[Category:Classical measure theory]]
5 KB (762 words) - 09:52, 16 August 2013
[[Category:Classical measure theory]]
A concept of classical measure theory.
5 KB (745 words) - 17:18, 18 August 2012
[[Category:Classical measure theory]]
...ed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
6 KB (884 words) - 09:36, 16 August 2013
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ame{supp} \phi_{j} \subset K$ for all $j$ and the coefficients of $\phi_j$ and all their derivatives tend uniformly to those of $\phi$.
5 KB (733 words) - 17:00, 1 July 2020
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.
11 KB (1,633 words) - 08:00, 6 June 2020
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
22 KB (3,436 words) - 22:12, 5 June 2020
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
4 KB (557 words) - 17:17, 7 January 2021
Please remove this comment and the {{TEX|auto}} line below,
be a self-adjoint and $ U $
3 KB (415 words) - 15:28, 28 February 2022
[[Category:Classical measure theory]]
.... Let $(X,\mu)$ be a space with a non-negative complete countably-additive measure $\mu$ (cf.
8 KB (1,363 words) - 05:26, 8 August 2018
Please remove this comment and the {{TEX|auto}} line below,
and $ b _ {i} $
8 KB (1,132 words) - 08:29, 6 June 2020
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
11 KB (1,718 words) - 22:15, 5 June 2020
[[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}}
10 KB (1,546 words) - 09:43, 16 August 2013
\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
3 KB (493 words) - 09:25, 16 August 2013
Please remove this comment and the {{TEX|auto}} line below,
where the integration is performed over the straight line $ \Pi _ \alpha $
7 KB (1,016 words) - 08:27, 6 June 2020
...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
6 KB (871 words) - 17:45, 1 July 2020
...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
3 KB (563 words) - 09:54, 27 November 2018
[[Category:Classical measure theory]]
...roperty of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional
9 KB (1,407 words) - 11:50, 4 February 2021
...1]</TD> <TD valign="top"> H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. 332 (Translated from G
...n="top">[a5]</TD> <TD valign="top"> M.M. Rao, "Stochastic processes and integration" , Sijthoff & Noordhoff (1979)</TD></TR>
2 KB (318 words) - 19:06, 11 October 2017
[[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]] $1$.
5 KB (734 words) - 11:57, 2 May 2014
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
2 KB (311 words) - 19:43, 5 June 2020
...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
2 KB (369 words) - 21:48, 1 January 2019
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
4 KB (631 words) - 08:46, 18 November 2012
[[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 $X$.
15 KB (2,431 words) - 07:07, 23 September 2012
Please remove this comment and the {{TEX|auto}} line below,
and 2) $ u( z ^ {0} + \lambda a) $
14 KB (2,065 words) - 08:06, 6 June 2020
[[Category:Classical measure theory]]
...that is, a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes]] probability measure);
14 KB (2,104 words) - 16:21, 17 April 2012
[[Category:Classical measure theory]]
...between the [[Measurable space|measurable spaces]] $(X, \mathcal{B} (X))$ and $(Y, \mathcal{B} (Y))$, see also [[Measurable mapping]].
5 KB (718 words) - 17:50, 5 May 2024
[[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]] $1$
7 KB (1,028 words) - 17:17, 18 August 2012
...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).
11 KB (1,738 words) - 18:15, 24 March 2018
...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
6 KB (1,043 words) - 14:24, 14 February 2020
[[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}}):
5 KB (795 words) - 09:26, 7 December 2012
...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
11 KB (1,653 words) - 17:46, 1 July 2020
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
8 KB (1,180 words) - 05:07, 23 February 2022
[[Category:Classical measure theory]]
...ure|signed measures]]. The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as:
9 KB (1,371 words) - 08:33, 16 August 2013
...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.
8 KB (1,117 words) - 20:03, 27 February 2021
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
finite measure $ \mu $
9 KB (1,287 words) - 16:56, 1 July 2020
[[Category:Classical measure theory]]
...\in E$ has [[Density of a set|density]] $1$ with respect to the [[Lebesgue measure]] $\lambda$, that is
6 KB (865 words) - 22:40, 17 August 2013
...$m$ 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;
7 KB (1,053 words) - 22:49, 7 March 2019
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
10 KB (1,460 words) - 16:45, 1 July 2020
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
21 KB (2,949 words) - 22:12, 5 June 2020
''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]].
46 KB (7,065 words) - 19:30, 1 January 2021
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]]], [[#
18 KB (2,598 words) - 22:11, 5 June 2020
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
6 KB (912 words) - 13:22, 28 October 2023
...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
7 KB (1,142 words) - 21:30, 1 January 2019
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} $
5 KB (645 words) - 21:34, 20 February 2021
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 $
7 KB (1,033 words) - 19:38, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
and with centre at the coordinate origin has the form
11 KB (1,592 words) - 12:54, 31 January 2022
Please remove this comment and the {{TEX|auto}} line below,
with respect to a measure $ \mu ( d \omega ) $
5 KB (678 words) - 06:03, 2 June 2023
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 $,
6 KB (919 words) - 18:23, 5 April 2023
...=x_0<x_1<\dots<x_n=b$ 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
6 KB (964 words) - 08:25, 25 April 2016
Please remove this comment and the {{TEX|auto}} line below,
is also left (right) almost-periodic, and the family $ f ( a x b ) $,
12 KB (1,716 words) - 11:05, 10 May 2020
Please remove this comment and the {{TEX|auto}} line below,
is a locally convex space if and only if the topology of $ E $
18 KB (2,674 words) - 22:17, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...n [[Orthonormal system|orthonormal system]] of functions with respect to a measure $ \mu $:
18 KB (2,612 words) - 13:13, 18 February 2022
Please remove this comment and the {{TEX|auto}} line below,
...y has been constructed for various spaces, primarily Euclidean, projective and homogeneous spaces.
28 KB (4,292 words) - 08:45, 1 July 2022
Please remove this comment and the {{TEX|auto}} line below,
is an orthogonal random measure on $ \Lambda $(
17 KB (2,406 words) - 20:00, 12 January 2024
...series, linear operators, random processes, and in the theory of extremal and approximation problems.
...www.encyclopediaofmath.org/legacyimages/h/h046/h046320/h04632022.png" />), and
37 KB (5,073 words) - 18:20, 1 December 2014
Please remove this comment and the {{TEX|auto}} line below,
with equality if and only if $ K $
6 KB (922 words) - 08:24, 26 March 2023
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 $
14 KB (2,252 words) - 08:09, 21 January 2024