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.
be a Banach space, and define a measurable function
, 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
In this case, the Bochner integral is defined by
These remarks show that the integral is well-defined (i.e independent of any choices).
Particularly useful is Bochner's criterion for integrability, which states that if
is a measure space, then a Bochner-measurable function
is called Bochner measurable if it is equal
-almost everywhere of a sequence of countably-valued simple functions.
is a continuous linear operator between Banach spaces
A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.
is a closed linear operator between Banach spaces
A version of the dominated convergence theorem also holds for the Bochner integral.
An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of ″nice″ Banach spaces.
which has bounded variation and is absolutely continuous with respect to
[2] Equivalent formulations include: It is known that the space
[2] 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, 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, statistical field theory ,and quantum field theory.
is Bochner integrable, its expectation is defined by
which inherits the usual linearity and continuity properties of the classical expectation.
, a stochastic process that is Banach-space valued.
The Bochner integral allows us to define the mean function
This approach is useful in stochastic partial differential equations, where each
is a random element in a functional space.
-valued random variables is called a martingale with respect to a filtration
The Bochner integral ensures that conditional expectations are well-defined in this Banach space setting.
The Bochner integral allows the definition of Gaussian measures on a Banach space, where one often encounters integrals of the form