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.

Definition

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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 integral

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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  

Properties

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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 theorem

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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:  

This is a consequence of the Hahn-Banach theorem and generalizes the mean value theorem for integrals of real-valued functions: If  , then closed convex sets are simply intervals and for  , the following inequalities hold:  

Existence

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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 bounded Cauchy 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 variables

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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]

See also

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References

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  • James 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 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
  • Sobolev, V. I. (2001) [1994], "Pettis integral", Encyclopedia of Mathematics, EMS Press