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
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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