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 where is a measure space and is a topological vector space (TVS) with a continuous dual space that separates points (i.e. if is nonzero then there is some such that ), e.g. is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing:

The map is called weakly measurable if for all , the scalar-valued map is a measurable map. A weakly measurable map is said to be weakly integrable on if there exists some such that for all , the scalar-valued map is Lebesgue integrable (that is, ) and

The map is said to be Pettis integrable if for all and also for every there exists a vector such that for all

In this case, we call the Pettis integral of on Common notations for the Pettis integral include

To understand the motivation behind the definition of "weakly integrable", consider the special case where is the underlying scalar field; that is, where or In this case, every linear functional on is of the form for some scalar (that is, is just scalar multiplication by a constant), the condition

simplifies to

In particular, in this special case, is weakly integrable on if and only if is Lebesgue integrable.

Properties

Mean value theorem

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

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

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

See also

References