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 $\mu[f 1_E]$.

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.

See also

• Vector measure
• Weakly measurable function

References

• J. K. Brooks, Representations of weak and strong integrals in Banach spaces, Proc. Nat. 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