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.

is a normed space or (more generally) is a Hausdorff locally convex TVS.

Evaluation of a functional may be written as a duality pairing:

is called weakly measurable if for all

A weakly measurable map

Common notations for the Pettis integral

To understand the motivation behind the definition of "weakly integrable", consider the special case where

is the underlying scalar field; that is, where

is just scalar multiplication by a constant), the condition

with the map scalar-valued functional on

is identified as a vector subspace of the double dual

is a semi-reflexive space if and only if this map is surjective.

An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If

for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms

The right-hand side is the lower Lebesgue integral of a

Taking a lower Lebesgue integral is necessary because the integrand

This follows from the Hahn-Banach theorem because for every vector

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 generalizes the mean value theorem for integrals of real-valued functions: If

, then closed convex sets are simply intervals and for

a Borel measure that assigns finite values to compact subsets,

is quasi-complete (that is, every bounded Cauchy net converges) and if

is weakly measurable and there exists a compact, convex

be a sequence of Pettis-integrable random variables, and write

denote the sample average.

Suppose that the partial sums

in the sense that all rearrangements of the sum converge to a single vector

The weak law of large numbers implies that

[citation needed] To get strong convergence, more assumptions are necessary.