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 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.

Definition

be a measure space be a Banach space. For generic Banach spaces, the Bochner integral extends the above cases.

and χE is the characteristic function then the simple function is integrable exactly as it is for the ordinary Lebesgue integral.

is Bochner integrable where the integral on the left-hand side is an ordinary Lebesgue integral.

In this case, the Bochner integral

is a Cauchy sequence.

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

 is called Bochner measurable-almost everywhere of a sequence of countably-valued simple functions.

Linear operators

.

A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators..

Dominated convergence theorem

A version of the dominated convergence theorem.

-valued vector measure which is absolutely continuous.

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.

if, for every countably-additive vector measure which has bounded variation

has the Radon–Nikodym property with respect to every finite measure. Equivalent formulations include:

• Bounded discrete-time martingales converge a.s.

• are differentiable a.e.

• has arbitrarily small diameter.

It is known that the space \ell_1 has the Radon–Nikodym property, but c_0 an infinite compact space, do not. Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem) and reflexive spaces, which include, in particular, Hilbert spaces.

Probability

The Bochner integral is used in probability theory 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, generalize 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, statistical field theory, and quantum field theory.

be a probability spaceand inherits the usual linearity and continuity properties of the classical expectation.

Stochastic process

, a stochastic process is a random element in a functional space.

In martingale theory-valued random variables is called a martingale with respect to a filtrationThe Bochner integral ensures that conditional expectations are well-defined in this Banach space setting.

Gaussian measure

The Bochner integral allows one to define moments for the Gaussian measure on a Banach space. If the Bochner integral exists, then it is equivalent to the Pettis integral denotes the dual pairing.

Extension to locally convex spaces

with values in some locally convex space (1975 Rybakov, 1981 Blondia, 2015 Beckmann and Deitmar). The extension by Beckmann and Deitmar uses the original approach of Bochner but generalized to netss and they distinguish three cases of assumptions on the locally convex space:

• is complete,

• is quasi-complete space is bounded,

• is finite.

Beckmann and Deitmar use the term of Bochner-approximability

They provide several equivalent characterizations of this property.

Blondia defines the extensions to locally convex spaces as follows

such that

• -almost surely

• , i.e.

• .

See also

References