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
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[[Category:Measure and integration]]
346 bytes (45 words) - 12:28, 9 November 2014
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...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
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5 KB (743 words) - 22:13, 5 June 2020
$#C+1 = 21 : ~/encyclopedia/old_files/data/F038/F.0308480 Feynman measure
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2 KB (349 words) - 19:39, 5 June 2020
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''measurable partition, of a [[Measure space|measure space]] $ ( M, \mu ) $''
5 KB (757 words) - 08:00, 6 June 2020
$#C+1 = 49 : ~/encyclopedia/old_files/data/Q076/Q.0706560 Quasi\AAhinvariant measure
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4 KB (564 words) - 17:20, 11 March 2021
[[Category:Set functions and measures on topological spaces]]
...mathcal{B}\to [0, \infty]$ 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
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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
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and $ u ( t, \omega ) $
8 KB (1,147 words) - 13:18, 26 March 2023
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..._ { 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
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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
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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
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...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
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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
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...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
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...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
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...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
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...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
$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
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7 KB (1,017 words) - 08:00, 6 June 2020
''generalized measure'', ''real valued measure'', ''charge''
[[Category:Classical measure theory]]
5 KB (762 words) - 09:52, 16 August 2013
[[Category:Classical measure theory]]
A concept of classical measure theory.
5 KB (745 words) - 17:18, 18 August 2012
[[Category:Classical measure theory]]
...ed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
6 KB (884 words) - 09:36, 16 August 2013
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...ame{supp} \phi_{j} \subset K$ 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
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...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
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...) 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
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...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