# Bochner integral

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.

## Definition

Let (*X*, Σ, μ) be a measure space and *B* a Banach space. The Bochner integral is defined in much the same way as the Lebesgue integral. First, a simple function is any finite sum of the form

where the *E*_{i} are disjoint members of the σ-algebra Σ, the *b*_{i} are distinct elements of *B*, and χ_{E} is the characteristic function of *E*. If *μ*(*E*_{i}) is finite whenever *b*_{i} ≠ 0, 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 ƒ : *X* → *B* is **Bochner integrable** if there exists a sequence of integrable simple functions *s*_{n} 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 a function is Bochner integrable if and only if it lies in the Bochner space .

## 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 (*X*, Σ, μ) is a measure space, then a Bochner-measurable function *ƒ* : *X* → *B* is Bochner integrable if and only if

A function *ƒ* : *X* → *B* is called Bochner-measurable if it is equal μ-almost everywhere to a function *g* taking values in a separable subspace *B*_{0} of *B*, and such that the inverse image *g*^{−1}(*U*) of every open set *U* in *B* belongs to Σ. Equivalently, *ƒ* is limit μ-almost everywhere of a sequence of simple functions.

If is a continuous linear operator, and is Bochner-integrable, then is Bochner-integrable and integration and may be interchanged:

This also holds for closed operators, given that be itself integrable (which, via the criterion mentioned above is trivially true for bounded ).

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if *ƒ*_{n} : *X* → *B* is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function *ƒ*, and if

for almost every *x* ∈ *X*, and *g* ∈ *L*^{1}(μ), then

as *n* → ∞ and

for all *E* ∈ Σ.

If *ƒ* is Bochner integrable, then the inequality

holds for all *E* ∈ Σ. In particular, the set function

defines a countably-additive *B*-valued vector measure on *X* which is absolutely continuous with respect to μ.

## Radon–Nikodym property

An important fact about the Bochner integral is that the Radon–Nikodym theorem *fails* to hold in general. This results in an important property of Banach spaces known as the Radon–Nikodym property. Specifically, if μ is a measure on (*X*, Σ), then *B* has the Radon–Nikodym property with respect to μ if, for every countably-additive vector measure
on (*X*, Σ) with values in *B* which has bounded variation and is absolutely continuous with respect to μ, there is a μ-integrable function *g* : *X* → *B* such that

for every measurable set *E* ∈ Σ.[1]

The Banach space *B* has the **Radon–Nikodym property** if *B* has the Radon–Nikodym property with respect to every finite measure. It is known that the space
has the Radon–Nikodym property, but
and the spaces
,
, for
an open bounded subset of
, and
, for *K* 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.

## See also

## References

- Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF).
*Divulgaciones Matemáticas*.**11**(1): 55–59 [pp. 55–56].

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