In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. 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.
Let where is a measure space and is a topological vector space (TVS) with a continuous dual space that separates points (that is, if is nonzero then there is some such that ), for example, is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing:
The map is called weakly measurable if for all the scalar-valued map is a measurable map. A weakly measurable map is said to be weakly integrable on if there exists some such that for all the scalar-valued map is Lebesgue integrable (that is, ) and
The map is said to be Pettis integrable if for all and also for every there exists a vector such that
In this case, is called the Pettis integral of on Common notations for the Pettis integral include
To understand the motivation behind the definition of "weakly integrable", consider the special case where is the underlying scalar field; that is, where or In this case, every linear functional on is of the form for some scalar (that is, is just scalar multiplication by a constant), the condition
simplifies to
In particular, in this special case, is weakly integrable on if and only if is Lebesgue integrable.
Relation to Dunford integraledit
The map is said to be Dunford integrable if for all and also for every there exists a vector called the Dunford integral of on such that
where
Identify every vector with the map scalar-valued functional on defined by This assignment induces a map called the canonical evaluation map and through it, is identified as a vector subspace of the double dual The space is a semi-reflexive space if and only if this map is surjective. The is Pettis integrable if and only if for every
Propertiesedit
An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If is linear and continuous and is Pettis integrable, then is Pettis integrable as well and
The standard estimate
for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms and all Pettis integrable ,
holds. The right-hand side is the lower Lebesgue integral of a -valued function, that is,
Taking a lower Lebesgue integral is necessary because the integrand may not be measurable. This follows from the Hahn-Banach theorem because for every vector there must be a continuous functional such that and for all , . Applying this to gives the result.
Mean value theoremedit
An important property is that the Pettis integral with respect to a finite measure is contained in the closure of the convex hull of the values scaled by the measure of the integration domain:
If is finite-dimensional then is Pettis integrable if and only if each of ’s coordinates is Lebesgue integrable.
If is Pettis integrable and is a measurable subset of , then by definition and are also Pettis integrable and
If is a topological space, its Borel--algebra, a Borel measure that assigns finite values to compact subsets, is quasi-complete (that is, every boundedCauchy net converges) and if is continuous with compact support, then is Pettis integrable. More generally: If is weakly measurable and there exists a compact, convex and a null set such that , then is Pettis-integrable.
Law of large numbers for Pettis-integrable random variablesedit
Let be a probability space, and let be a topological vector space with a dual space that separates points. Let be a sequence of Pettis-integrable random variables, and write for the Pettis integral of (over ). Note that is a (non-random) vector in and is not a scalar value.
Let
denote the sample average. By linearity, is Pettis integrable, and
Suppose that the partial sums
converge absolutely in the topology of in the sense that all rearrangements of the sum converge to a single vector The weak law of large numbers implies that for every functional Consequently, in the weak topology on
Without further assumptions, it is possible that does not converge to [citation needed] To get strong convergence, more assumptions are necessary.[citation needed]
Israel 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 Zbl 0014.16202
Michel Talagrand, Pettis Integral and Measure Theory, Memoirs of the AMS no. 307 (1984) MR0756174
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In mathematics the Pettis integral or Gelfand Pettis integral named after Israel M Gelfand and Billy James Pettis extends the definition of the Lebesgue integral to vector valued functions on a measure space by exploiting duality 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 Contents 1 Definition 2 Relation to Dunford integral 3 Properties 3 1 Mean value theorem 3 2 Existence 4 Law of large numbers for Pettis integrable random variables 5 See also 6 ReferencesDefinition editLet f X V displaystyle f X to V nbsp where X S m displaystyle X Sigma mu nbsp is a measure space and V displaystyle V nbsp is a topological vector space TVS with a continuous dual space V displaystyle V nbsp that separates points that is if x V displaystyle x in V nbsp is nonzero then there is some l V displaystyle l in V nbsp such that l x 0 displaystyle l x neq 0 nbsp for example V displaystyle V nbsp is a normed space or more generally is a Hausdorff locally convex TVS Evaluation of a functional may be written as a duality pairing f x f x displaystyle langle varphi x rangle varphi x nbsp The map f X V displaystyle f X to V nbsp is called weakly measurable if for all f V displaystyle varphi in V nbsp the scalar valued map f f displaystyle varphi circ f nbsp is a measurable map A weakly measurable map f X V displaystyle f X to V nbsp is said to be weakly integrable on X displaystyle X nbsp if there exists some e V displaystyle e in V nbsp such that for all f V displaystyle varphi in V nbsp the scalar valued map f f displaystyle varphi circ f nbsp is Lebesgue integrable that is f f L1 X S m displaystyle varphi circ f in L 1 left X Sigma mu right nbsp andf e Xf f x dm x displaystyle varphi e int X varphi f x mathrm d mu x nbsp The map f X V displaystyle f X to V nbsp is said to be Pettis integrable if f f L1 X S m displaystyle varphi circ f in L 1 left X Sigma mu right nbsp for all f V displaystyle varphi in V prime nbsp and also for every A S displaystyle A in Sigma nbsp there exists a vector eA V displaystyle e A in V nbsp such that f eA A f f x dm x for all f V displaystyle langle varphi e A rangle int A langle varphi f x rangle mathrm d mu x quad text for all varphi in V nbsp In this case eA displaystyle e A nbsp is called the Pettis integral of f displaystyle f nbsp on A displaystyle A nbsp Common notations for the Pettis integral eA displaystyle e A nbsp include Afdm Af x dm x and in case that A X is understood m f displaystyle int A f mathrm d mu qquad int A f x mathrm d mu x quad text and in case that A X text is understood quad mu f nbsp To understand the motivation behind the definition of weakly integrable consider the special case where V displaystyle V nbsp is the underlying scalar field that is where V R displaystyle V mathbb R nbsp or V C displaystyle V mathbb C nbsp In this case every linear functional f displaystyle varphi nbsp on V displaystyle V nbsp is of the form f y sy displaystyle varphi y sy nbsp for some scalar s V displaystyle s in V nbsp that is f displaystyle varphi nbsp is just scalar multiplication by a constant the conditionf e Af f x dm x for all f V displaystyle varphi e int A varphi f x mathrm d mu x quad text for all varphi in V nbsp simplifies to se Asf x dm x for all scalars s displaystyle se int A sf x mathrm d mu x quad text for all scalars s nbsp In particular in this special case f displaystyle f nbsp is weakly integrable on X displaystyle X nbsp if and only if f displaystyle f nbsp is Lebesgue integrable Relation to Dunford integral editThe map f X V displaystyle f X to V nbsp is said to be Dunford integrable if f f L1 X S m displaystyle varphi circ f in L 1 left X Sigma mu right nbsp for all f V displaystyle varphi in V prime nbsp and also for every A S displaystyle A in Sigma nbsp there exists a vector dA V displaystyle d A in V nbsp called the Dunford integral of f displaystyle f nbsp on A displaystyle A nbsp such that dA f A f f x dm x for all f V displaystyle langle d A varphi rangle int A langle varphi f x rangle mathrm d mu x quad text for all varphi in V nbsp where dA f dA f displaystyle langle d A varphi rangle d A varphi nbsp Identify every vector x V displaystyle x in V nbsp with the map scalar valued functional on V displaystyle V nbsp defined by f V f x displaystyle varphi in V mapsto varphi x nbsp This assignment induces a map called the canonical evaluation map and through it V displaystyle V nbsp is identified as a vector subspace of the double dual V displaystyle V nbsp The space V displaystyle V nbsp is a semi reflexive space if and only if this map is surjective The f X V displaystyle f X to V nbsp is Pettis integrable if and only if dA V displaystyle d A in V nbsp for every A S displaystyle A in Sigma nbsp Properties editAn immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators If F V1 V2 displaystyle Phi colon V 1 to V 2 nbsp is linear and continuous and f X V1 displaystyle f colon X to V 1 nbsp is Pettis integrable then F f displaystyle Phi circ f nbsp is Pettis integrable as well and XF f x dm x F Xf x dm x displaystyle int X Phi f x d mu x Phi left int X f x d mu x right nbsp The standard estimate Xf x dm x X f x dm x displaystyle left int X f x d mu x right leq int X f x d mu x nbsp for real and complex valued functions generalises to Pettis integrals in the following sense For all continuous seminorms p V R displaystyle p colon V to mathbb R nbsp and all Pettis integrable f X V displaystyle f colon X to V nbsp p Xf x dm x X p f x dm x displaystyle p left int X f x d mu x right leq underline int X p f x d mu x nbsp holds The right hand side is the lower Lebesgue integral of a 0 displaystyle 0 infty nbsp valued function that is X gdm sup Xhdm h X 0 is measurable and 0 h g displaystyle underline int X g d mu sup left left int X h d mu right h colon X to 0 infty text is measurable and 0 leq h leq g right nbsp Taking a lower Lebesgue integral is necessary because the integrand p f displaystyle p circ f nbsp may not be measurable This follows from the Hahn Banach theorem because for every vector v V displaystyle v in V nbsp there must be a continuous functional f V displaystyle varphi in V nbsp such that f v p v displaystyle varphi v p v nbsp and for all w V displaystyle w in V nbsp f w p w displaystyle varphi w leq p w nbsp Applying this to v Xfdm displaystyle v int X f d mu nbsp gives the result Mean value theorem edit An important property is that the Pettis integral with respect to a finite measure is contained in the closure of the convex hull of the values scaled by the measure of the integration domain m A lt implies Afdm m A co f A displaystyle mu A lt infty text implies int A f d mu in mu A cdot overline operatorname co f A nbsp This is a consequence of the Hahn Banach theorem and generalizes the mean value theorem for integrals of real valued functions If V R displaystyle V mathbb R nbsp then closed convex sets are simply intervals and for f X a b displaystyle f colon X to a b nbsp the following inequalities hold m A a Afdm m A b displaystyle mu A a leq int A f d mu leq mu A b nbsp Existence edit If V Rn displaystyle V mathbb R n nbsp is finite dimensional then f displaystyle f nbsp is Pettis integrable if and only if each of f displaystyle f nbsp s coordinates is Lebesgue integrable If f displaystyle f nbsp is Pettis integrable and A S displaystyle A in Sigma nbsp is a measurable subset of X displaystyle X nbsp then by definition f A A V displaystyle f A colon A to V nbsp and f 1A X V displaystyle f cdot 1 A colon X to V nbsp are also Pettis integrable and Af Adm Xf 1Adm displaystyle int A f A d mu int X f cdot 1 A d mu nbsp If X displaystyle X nbsp is a topological space S BX displaystyle Sigma mathfrak B X nbsp its Borel s displaystyle sigma nbsp algebra m displaystyle mu nbsp a Borel measure that assigns finite values to compact subsets V displaystyle V nbsp is quasi complete that is every bounded Cauchy net converges and if f displaystyle f nbsp is continuous with compact support then f displaystyle f nbsp is Pettis integrable More generally If f displaystyle f nbsp is weakly measurable and there exists a compact convex C V displaystyle C subseteq V nbsp and a null set N X displaystyle N subseteq X nbsp such that f X N C displaystyle f X setminus N subseteq C nbsp then f displaystyle f nbsp is Pettis integrable Law of large numbers for Pettis integrable random variables editLet W F P displaystyle Omega mathcal F operatorname P nbsp be a probability space and let V displaystyle V nbsp be a topological vector space with a dual space that separates points Let vn W V displaystyle v n Omega to V nbsp be a sequence of Pettis integrable random variables and write E vn displaystyle operatorname E v n nbsp for the Pettis integral of vn displaystyle v n nbsp over X displaystyle X nbsp Note that E vn displaystyle operatorname E v n nbsp is a non random vector in V displaystyle V nbsp and is not a scalar value Letv N 1N n 1Nvn displaystyle bar v N frac 1 N sum n 1 N v n nbsp denote the sample average By linearity v N displaystyle bar v N nbsp is Pettis integrable and E v N 1N n 1NE vn V displaystyle operatorname E bar v N frac 1 N sum n 1 N operatorname E v n in V nbsp Suppose that the partial sums1N n 1NE v n displaystyle frac 1 N sum n 1 N operatorname E bar v n nbsp converge absolutely in the topology of V displaystyle V nbsp in the sense that all rearrangements of the sum converge to a single vector l V displaystyle lambda in V nbsp The weak law of large numbers implies that f E v N l 0 displaystyle langle varphi operatorname E bar v N lambda rangle to 0 nbsp for every functional f V displaystyle varphi in V nbsp Consequently E v N l displaystyle operatorname E bar v N to lambda nbsp in the weak topology on X displaystyle X nbsp Without further assumptions it is possible that E v N displaystyle operatorname E bar v N nbsp does not converge to l displaystyle lambda nbsp citation needed To get strong convergence more assumptions are necessary citation needed See also editBochner measurable function Bochner integral Bochner space Type of topological space Vector measure Weakly measurable functionReferences editJames K Brooks Representations of weak and strong integrals in Banach spaces Proceedings of the National Academy of Sciences of the United States of America 63 1969 266 270 Fulltext MR0274697 Israel M Gel fand Sur un lemme de la theorie des espaces lineaires Commun Inst Sci Math et Mecan Univ Kharkoff et Soc Math Kharkoff IV Ser 13 1936 35 40 Zbl 0014 16202 Michel Talagrand Pettis Integral and Measure Theory Memoirs of the AMS no 307 1984 MR0756174 Sobolev V I 2001 1994 Pettis integral Encyclopedia of Mathematics EMS Press Retrieved from https en wikipedia org w index php title Pettis 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