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 $f\colon X\to V$ , where $(X,\Sigma,\mu)$ is a measure space and $V$ is a topological vector space. Suppose that $V$ admits a dual space $V^{*}$ that separates points. e.g., $V$ a Banach space or (more generally) a locally convex, Hausdorff vector space. We write evaluation of a functional as duality pairing: $\langle \varphi ,x\rangle =\varphi [x]$ .

Choose any measurable set $E\in \Sigma$ . We say that $f$ is Pettis integrable (over $E$ ) if there exists a vector $e\in V$ so that

\langle \varphi ,e\rangle =\int _{E}\langle \varphi ,f(x)\rangle \,d\mu (x){\text{ for all functionals }}\varphi \in V^{*}.

In this case, we call $e$ the Pettis integral of $f$ (over $E$ ). Common notations for the Pettis integral $e$ include $\int _{E}f\mu$ , $\int _{E}f(t)\,d\mu (t)$ and $\mu [f1_{E}]$ . Note that if $V=\mathbb {R} ^{n}$ is finite-dimensional then $f$ is Pettis integrable over $E$ if and only if each of $f$ 's coordinates is integrable over $E$ .

A function is Pettis integrable (over $X$ ) if the scalar-valued function $\varphi \circ f$ is integrable for every functional $\varphi \in X^{*}$ .

Law of Large Numbers for Pettis integrable random variables

Let $(\Omega ,{\mathcal F},{\mathbb P})$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_{n}:\Omega \to V$ be a sequence of Pettis integrable random variables, and write ${\mathbb E}[v_{n}]$ for the Pettis integral of $v_{n}$ (over $X$ ). Note that ${\mathbb E}[v_{n}]$ is a (non-random) vector in $V$ , and is not a scalar value.

Let ${\bar v}_{N}:={\frac {1}{N}}\sum _{{n=1}}^{N}v_{n}$ denote the sample average. By linearity, ${\bar v}_{N}$ is Pettis integrable, and ${\mathbb E}[{\bar v}_{N}]={\frac {1}{N}}\sum _{{n=1}}^{N}{\mathbb E}[v_{n}]$ in $V$ .

Suppose that the partial sums ${\frac {1}{N}}\sum _{{n=1}}^{N}{\mathbb E}[{\bar v}_{n}]$ converge absolutely in the topology of $V$ , in the sense that all rearrangements of the sum converge to a single vector $\lambda \in V$ . The Weak Law of Large Numbers implies that $\langle \varphi ,{\mathbb E}[{\bar v}_{N}]-\lambda \rangle \to 0$ for every functional $\varphi \in V^{*}$ . Consequently, ${\mathbb E}[{\bar v}_{N}]\to \lambda$ in the weak topology on $X$ .

Without further assumptions, it is possible that ${\mathbb E}[{\bar v}_{N}]$ does not converge to $\lambda$ . To get strong convergence, more assumptions are necessary.

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

Integrals

Numerical integration	Riemann integral Lebesgue integral Burkill integral Bochner integral Daniell integral Darboux integral Henstock–Kurzweil integral Haar integral Hellinger integral Khinchin integral Kolmogorov integral Lebesgue–Stieltjes integral Pettis integral Pfeffer integral Riemann–Stieltjes integral Regulated integral

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Pettis integral

Definition

Law of Large Numbers for Pettis integrable random variables

See also

References