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 Bochner integral, named for
Salomon Bochner
Salomon Bochner (20 August 1899 – 2 May 1982) was a Galizien-born mathematician, known for work in mathematical analysis, probability theory and differential geometry.
Life
He was born into a Jewish family in Podgórze (near Kraków), th ...
, extends the definition of a multidimensional
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 functions that take values in a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
, as the limit of integrals of
simple function
In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reas ...
s.
The Bochner integral provides the mathematical foundation for extensions of basic
integral transforms into more abstract spaces, vector-valued functions, and operator spaces. Examples of such extensions include vector-valued
Laplace transforms and abstract
Fourier transforms
In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the origin ...
.
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 (, ) 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 vectors and ...
, and define a measurable function
. When
, we have the standard Lebesgue integral
, and when
, we have the standard multidimensional Lebesgue integral
. For generic Banach spaces, the Bochner integral extends the above cases.
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 is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...
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
In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a Banach space which is not necessarily the space \R or \Complex of real or complex numbers.
The space L^p(X) consists of (equivalen ...
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 countably-valued 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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
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
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship betwe ...
with respect to
.
Radon–Nikodym property
An important fact about the Bochner integral is that the
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
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
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
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 for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
s, which include, in particular,
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
s.
Probability
The Bochner integral is used in
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
for handling random variables and stochastic processes that take values in a Banach space. The classical convergence theorems—such as the dominated convergence theorem—when applied to the Bochner integral, generalizes laws of large numbers and central limit theorems for sequences of Banach-space valued random variables. Such integrals are central to the analysis of distributions in functional spaces and have applications in fields such as
stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
,
statistical field theory
In theoretical physics, statistical field theory (SFT) is a theoretical framework that describes phase transitions. It does not denote a single theory but encompasses many models, including for magnetism, superconductivity, superfluidity, topologi ...
, and
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.
Let
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
, and consider a random variable
taking values in a Banach space
. When
is Bochner integrable, its expectation is defined by
which inherits the usual linearity and continuity properties of the classical expectation.
Stochastic process
Consider
, a stochastic process that is Banach-space valued. The Bochner integral allows us to define the mean function
whenever each
is Bochner integrable. This approach is useful in stochastic partial differential equations, where each
is a random element in a functional space.
In
martingale theory
In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given ...
, a sequence
of
-valued random variables is called a martingale with respect to a
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
if each
is
-measurable, Bochner integrable, and satisfies
The Bochner integral ensures that conditional expectations are well-defined in this Banach space setting.
Gaussian measure
The Bochner integral allows the definition of
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
s on a Banach space, where one often encounters integrals of the form
where
and
denotes the
dual pairing.
See also
*
*
*
*
*
References
*
*
*
*
*
*
*
*
*
*
{{Analysis in topological vector spaces
Definitions of mathematical integration
Integral representations
Topological vector spaces