In
mathematics, the Pettis integral or Gelfand–Pettis integral, named after
Israel M. Gelfand and
Billy James Pettis
Billy James Pettis (1913 – 14 April 1979), was an American mathematician, known for his contributions to functional analysis.
See also
* Dunford–Pettis property
* Dunford–Pettis theorem
*Milman–Pettis theorem
*Orlicz–Pettis theorem
*Pet ...
, extends the definition of the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
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 In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.
Definition
Let (X, \Sigma, \mu) be a me ...
, 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 or (more generally) is a Hausdorff
locally convex TVS.
Evaluation of a functional may be written as a
duality pairing
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
:
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 of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
(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 if and only if this map is
surjective.
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 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