Pettis integral


In mathematics, the Pettis integral or Gelfand-Pettis integral, named after Israel M. Gelfand and Billy James 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

Let f : X → V where is a measure space and V is a topological vector space with a continuous dual space that separates points, e.g. V is a normed space or is a Hausdorff locally convex TVS.
We write evaluation of a functional as duality pairing:.
We say that f is Pettis integrable if for all and there exists a vector e ∈ V so that:
In this case, we call e the Pettis integral of f. Common notations for the Pettis integral include

Properties

An important property is that the Pettis integral with respect to 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

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 . Note that is a 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. To get strong convergence, more assumptions are necessary.