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.
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.
- 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