Let where is a measure space and is a topological vector space. Suppose that admits a dual space that separates points, e.g. is a Banach space or (more generally) is a locally-convex Hausdorff vector space. We write evaluation of a functional as duality pairing: .
We say that is Pettis integrable if for all and there exists a vector so that:
In this case, we call the Pettis integral of . Common notations for the Pettis integral include
An immediate consequence of the definition is that Pettis integrals are compatible with continuous, linear operators: If is and linear and continuous and is Pettis integrable, then is Pettis integrable as well and:
The standard estimate
for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms and all Pettis integrable
holds. The right hand side is the lower Lebesgue integral of a -valued function, i.e.
Taking a lower Lebesgue integral is necessary because the integrand may not be measurable. This follows from the Hahn-Banach theorem because for every vector there must be a continuous functional such that and . Applying this to it gives the result.
More generally: If is weakly measurable and there exists a compact, convex and a null set such that , then is Pettis-integrable.
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.
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.