# Pettis integral

In mathematics, the **Pettis integral** or **Gelfand–Pettis integral**, named after I. M. Gelfand and B. J. 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

## Definition[edit]

Let where is a measure space and is a topological vector space. Suppose that admits a dual space that separates points, e.g. is a Banach space or (more generally) a is locally-convex Hausdorff vector space. We write evaluation of a functional as duality pairing: .

We say that is Pettis integrable if for all and there exists a vector so that:

In this case, we call the Pettis integral of . Common notations for the Pettis integral include

## Properties[edit]

- An immediate consequence of the definition is that Pettis integrals are compatible with continuous, linear operators: If is and linear and continuous and is Pettis integrable, then is Pettis integrable as well and holds.
- 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. Here denotes the lower Lebesgue integral of a -valued function, i.e. . Taking an 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 . Applying this to it gives the result.

### Mean value theorem[edit]

An important property is that the Pettis integral w.r.t. 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 generalises the mean value theorem for integrals of real-valued functions: If , then closed convex sets are simply intervals and for , the inequalities hold.

### Existence[edit]

- 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 and are also Pettis integrable and holds.
- If is a topological space, its Borel--algebra, a Borel measure that assigns finite values to compact subsets, is quasi-complete (i.e. if every
*bounded*Cauchy net converges) and if is continuous with compact support, then is Pettis integrable. - More generally: If is weakly measurable, there exists a compact, convex and a null set such that , then is Pettis-integrable.

## Law of large numbers for Pettis-integrable random variables[edit]

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[edit]

## References[edit]

- J. K. Brooks,
*Representations of weak and strong integrals in Banach spaces*, Proc. Natl. Acad. Sci. U.S.A. 63, 1969, 266–270. Fulltext MR0274697 - 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 Zbl 0014.16202 - M. Talagrand,
*Pettis Integral and Measure Theory*, Memoirs of the AMS no. 307 (1984) MR0756174 - Sobolev, V. I. (2001) [1994], "Pettis integral", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4