# 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.

## Definition

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

Choose any measurable set . We say that is Pettis integrable (over ) if there exists a vector so that

In this case, we call the Pettis integral of (over ). Common notations for the Pettis integral include , and . Note that if is finite-dimensional then is Pettis integrable over if and only if each of 's coordinates is integrable over .

A function is Pettis integrable (over ) if the scalar-valued function is integrable for every functional .

## Law of Large Numbers for Pettis integrable random variables

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 in .

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 . To get strong convergence, more assumptions are necessary.

## See also

## References

- J. K. Brooks,
*Representations of weak and strong integrals in Banach spaces*, Proc. Natl. Acad. Sci. U.S.A. 63, 1969, 266–270. Fulltext MR 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 Zbl 0014.16202 - M. Talagrand,
*Pettis Integral and Measure Theory*, Memoirs of the AMS no. 307 (1984) MR 0756174 - Sobolev, V. I. (2001), "Pettis integral", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4