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.
is defined in much the same way as the Lebesgue integral.
First, define a simple function to be any finite sum of the form
and χE is the characteristic function of
exactly as it is for the ordinary Lebesgue integral.
In this case, the Bochner integral is defined by
is a Cauchy sequence 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).
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 called Bochner measurable if it is equal
taking values in a separable subspace
-almost everywhere of a sequence of countably-valued simple functions.
is a continuous linear operator between Banach spaces
is Bochner integrable, then it is relatively straightforward to show that
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.
is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function
is Bochner integrable, then the inequality
which is absolutely continuous with respect to
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.
if, for every countably-additive vector measure
which has bounded variation and is absolutely continuous with respect to
has the Radon–Nikodym property with respect to every finite measure.
[2] Equivalent formulations include: It is known that the space
an open bounded subset of
an infinite compact space, do not.