In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Bochner integral, named for
Salomon Bochner
Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry.
Life
He was born into a Jewish family in Podgórze (near Kraków), then ...
, extends the definition of
Lebesgue integral to functions that take values in a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, as the limit of integrals of
simple functions.
Definition
Let
be 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 ...
, and
be a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. The Bochner integral of a function
is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form
where the
are disjoint members of the
-algebra
the
are distinct elements of
and χ
E is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of
If
is finite whenever
then the simple function is integrable, and the integral is then defined by
exactly as it is for the ordinary Lebesgue integral.
A measurable function
is Bochner integrable if there exists a sequence of integrable simple functions
such that
where the integral on the left-hand side is an ordinary Lebesgue integral.
In this case, the Bochner integral is defined by
It can be shown that the sequence
is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in the Banach space
hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions
These remarks show that the integral is well-defined (i.e independent of any choices). It can be shown that a function is Bochner integrable if and only if it lies in the
Bochner space
Properties
Elementary properties
Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Particularly useful is Bochner's criterion for integrability, which states that if
is a measure space, then a Bochner-measurable function
is Bochner integrable if and only if
Here, a function
is called Bochner measurable if it is equal
-almost everywhere to a function
taking values in a separable subspace
of
, and such that the inverse image
of every open set
in
belongs to
. Equivalently,
is the limit
-almost everywhere of a sequence of simple functions.
Linear operators
If
is a continuous linear operator between Banach spaces
and
, and
is Bochner integrable, then it is relatively straightforward to show that
is Bochner integrable and integration and the application of
may be interchanged:
for all measurable subsets
.
A non-trivially stronger form of this result, known as Hille's theorem, also holds for
closed operators. If
is a closed linear operator between Banach spaces
and
and both
and
are Bochner integrable, then
for all measurable subsets
.
Dominated convergence theorem
A version of the
dominated convergence theorem also holds for the Bochner integral. Specifically, if
is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function
, and if
for almost every
, and
, then
as
and
for all
.
If
is Bochner integrable, then the inequality
holds for all
In particular, the set function
defines a countably-additive
-valued
vector measure on
which is
absolutely continuous with respect to
.
Radon–Nikodym property
An important fact about the Bochner integral is that the
Radon–Nikodym theorem to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of nice Banach spaces.
Specifically, if
is a measure on
then
has the Radon–Nikodym property with respect to
if, for every countably-additive
vector measure on
with values in
which has
bounded variation and is absolutely continuous with respect to
there is a
-integrable function
such that
for every measurable set
The Banach space
has the Radon–Nikodym property if
has the Radon–Nikodym property with respect to every finite measure.
Equivalent formulations include:
* Bounded discrete-time
martingales in
converge a.s.
[. Thm. 2.3.6-7, conditions (1,4,10).]
* Functions of bounded-variation into
are differentiable a.e.
* For every bounded
, there exists
and
such that
has arbitrarily small diameter.
It is known that the space
has the Radon–Nikodym property, but
and the spaces
for
an open bounded subset of
and
for
an infinite compact space, do not.
[.] Spaces with Radon–Nikodym property include separable dual spaces (this is the
Dunford–Pettis theorem) and
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an i ...
s, which include, in particular,
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.
See also
*
*
*
*
*
References
*
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{{Analysis in topological vector spaces
Definitions of mathematical integration
Integral representations
Topological vector spaces