Čech-to-derived functor spectral sequence: Difference between revisions

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In [[mathematics]], the '''Pettis integral''' or '''Gelfand–Pettis integral''', named after [[Israel Gelfand|I. M. Gelfand]] and [[Billy James Pettis|B. J. Pettis]], extends the definition of the [[Lebesgue integral]] to vector-valued functions on a [[measure space]], by exploiting [[dual space|duality]].  
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The integral was introduced by Gelfand for the case when the measure space is an interval with [[Lebesgue measure]]. The integral is also called the '''weak integral''' in contrast to the [[Bochner integral]], which is the strong integral.
 
==Definition==
 
Suppose that <math>f\colon X\to V</math>, where <math>(X,\Sigma,\mu)</math> is a measure space and <math>V</math> is a topological vector space. Suppose that <math>V</math> admits a dual space <math>V^*</math> that separates points. e.g., <math>V</math> a Banach space or (more generally) a locally convex, Hausdorff vector space. We write evaluation of a functional as duality pairing: <math>\langle \varphi, x \rangle = \varphi[x]</math>.
 
Choose any measurable set <math>E \in \Sigma</math> We say that <math>f</math> is Pettis integrable (over <math>E</math>) if there exists a vector <math>e \in V</math> so that
: <math> \langle \varphi, e\rangle = \int_E \langle \varphi, f(x) \rangle \, d\mu(x)\text{ for all functionals }\varphi\in V^*.</math>
In this case, we call <math>e</math> the Pettis integral of <math>f</math> (over <math>E</math>). Common notations for the Pettis integral <math>e</math> include <math>\int_E f \mu</math>, <math>\int_E f(t) \, d\mu(t)</math> and <math>\mu[f 1_E]</math>.
 
A function is Pettis integrable (over <math>X</math>) if the scalar-valued function <math>\varphi \circ f</math> is integrable for every functional <math>\varphi \in X^*</math>.
 
==Law of Large Numbers for Pettis integrable random variables==
 
Let <math>(\Omega, \mathcal F, \mathbb P)</math> be a probability space, and let <math>V</math> be a topological vector space with a dual space that separates points. Let <math>v_n : \Omega \to V</math> be a sequence of Pettis integrable random variables, and write <math>\mathbb E[v_n]</math> for the Pettis integral of <math>v_n</math> (over <math>X</math>). Note that <math>\mathbb E[v_n]</math> is a (non-random) vector in <math>V</math>, and is not a scalar value.
 
Let <math>\bar v_N := \frac{1}{N} \sum_{n=1}^N v_n</math> denote the sample average. By linearity, <math>\bar v_N</math> is Pettis integrable, and <math>\mathbb E[\bar v_N] = \frac{1}{N} \sum_{n=1}^N \mathbb E[v_n]</math> in <math>V</math>.
 
Suppose that the partial sums <math>\frac{1}{N} \sum_{n=1}^N \mathbb E[\bar v_n]</math> converge absolutely in the topology of <math>V</math>, in the sense that all rearrangements of the sum converge to a single vector <math>\lambda \in V</math>. The Weak Law of Large Numbers implies that <math>\langle \varphi, \mathbb E[\bar v_N] - \lambda \rangle \to 0</math> for every functional <math>\varphi \in V^*</math>. Consequently, <math>\mathbb E[\bar v_N] \to \lambda</math> in the [[weak topology]] on <math>X</math>.
 
Without further assumptions, it is possible that <math>\mathbb E[\bar v_N]</math> does not converge to <math>\lambda</math>.{{Citation needed|date=October 2013}} To get strong convergence, more assumptions are necessary.{{Citation needed|date=October 2013}}
 
==See also==
* [[Vector measure]]
* [[Weakly measurable function]]
 
==References==
* J. K. Brooks, ''Representations of weak and strong integrals in Banach spaces'',  Proc. Nat. Acad. Sci. U.S.A. 63, 1969, 266–270. [http://www.pnas.org/content/63/2/266.abstract Fulltext] {{MR|id=0274697}}
*  I.M. Gel'fand, ''Sur un lemme de la théorie des espaces linéaires'', Commun. Inst. Sci. Math. et Mecan., Univ. Kharkoff et Soc. Math. Kharkoff, IV. Ser. 13, 1936, 35–40 [http://www.zentralblatt-math.org/zmath/en/search/?q=an:0014.16202 Zbl 0014.16202]
* [[Michel Talagrand|M. Talagrand]], ''Pettis Integral and Measure Theory'', Memoirs of the AMS no. 307 (1984) {{MR|id=0756174}}
* {{springer|title=Pettis integral|id=p/p072490|first=V. I.|last=Sobolev}}
 
{{integral}}
{{Functional Analysis}}
 
[[Category:Functional analysis]]
[[Category:Integrals]]

Latest revision as of 16:13, 26 November 2014

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