In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Pettis integral or Gelfand–Pettis integral, named after
Israel M. Gelfand and
Billy James Pettis, extends the definition of the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
to vector-valued functions on a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, by exploiting
duality. The integral was introduced by Gelfand for the case when the measure space is an interval with
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. The integral is also called the weak integral in contrast to the
Bochner integral
In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of a multidimensional Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.
The Bochner integral p ...
, which is the strong integral.
Definition
Let
where
is a measure space and
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) with a continuous dual space
that separates points (that is, if
is nonzero then there is some
such that
), for example,
is a
normed space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
or (more generally) is a Hausdorff
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
TVS.
Evaluation of a functional may be written as a
duality pairing
In mathematics, a dual system, dual pair or a duality over a Field (mathematics), field \mathbb is a triple (X, Y, b) consisting of two vector spaces, X and Y, over \mathbb and a non-Degenerate bilinear form, degenerate bilinear map b : X \times Y ...
:
The map
is called if for all
the scalar-valued map
is a
measurable map.
A weakly measurable map
is said to be if there exists some
such that for all
the scalar-valued map
is
Lebesgue integrable
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
(that is,
) and
The map
is said to be if
for all
and also for every
there exists a vector
such that
In this case,
is called the 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.
Relation to Dunford integral
The map
is said to be if
for all
and also for every
there exists a vector
called the of
on
such that
where
Identify every vector
with the map scalar-valued functional on
defined by
This assignment induces a map called the canonical evaluation map and through it,
is identified as a vector subspace of the double dual
The space
is a
semi-reflexive space In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of ''X'') is bijecti ...
if and only if this map is
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
.
The
is Pettis integrable if and only if
for every
Properties
An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If
is 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