Bochner integral

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In mathematics, the Bochner integral extends the definition of Lebesgue integral to functions which take values in a Banach space.

The theory of vector-valued functions is a chapter of mathematical analysis, concerned with the generalisation to functions taking values in a Banach space, or more general topological vector space, of the notions of infinite sum and integral. It includes as a particular case the idea of operator-valued function, basic in spectral theory, and this case provided much of the motivation around 1930. When the vectors lie in a space of finite dimension, everything typically can be done component-by-component.

Infinite sums of vectors in a Banach space B, which is a fortiori a complete metric space, converge just when they are Cauchy sequences with respect to the norm of the space. This case, of functions from the natural numbers to B, presents no particular fresh difficulty. However, some new difficulties arise when considering the integral of functions from a general measure space into a Banach space. These difficulties may be addressed by a straightforward generalization of the usual approach to the Lebesgue integral via simple functions. An integral of a vector-valued function with respect to a measure is often called a Bochner integral, for Salomon Bochner.

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[edit] 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

s(x) = \sum_{i=1}^n \chi_{E_i}(x) b_i

where the Ei are disjoint members of the σ-algebra Σ, the bi are distinct elements of B, and χE is the characteristic function of E. The integral of a simple function is then defined by

\int_X \left[\sum_{i=1}^n \chi_{E_i}(x) b_i\right]\, d\mu = \sum_{i=1}^n \mu(E_i) b_i

exactly as it is for the ordinary Lebesgue integral.

A measurable function ƒ : X → B is Bochner integrable if there exists a sequence sn of simple functions such that

\lim_{n\to\infty}\int_X \|f-s_n\|_B\,d\mu = 0,

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

In this case, the Bochner integral is defined by

\int_X f\, dx = \lim_{n\to\infty}\int_X s_n.

[edit] Properties

Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Perhaps the most striking example is Bochner's criterion for integrability, which states that if (X,Σ,μ) is a finite measure space, then a measurable function f : X → B is Bochner integrable if and only if

\int_X \|f\|_B\, d\mu < \infty.

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if fn : X → B is a sequence of measurable functions tending almost everywhere to a limit function f, and if

\|f_n(x)\|_B\le g(x)

for almost every x∈X, and g∈L1(μ), then

\int_X \|f-f_n\|_B\,d\mu \to 0

as n→∞ and

\int_E f_n\,d\mu \to \int_E f\,d\mu

for all E∈Σ.

If f is Bochner integrable, then the inequality

\left\|\int_Ef\,d\mu\right\|_B \le \int_E \|f\|_B\,d\mu

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

E\mapsto \int_E f\, d\mu

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

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 the Banach space B, then B has the Radon-Nikodym property with respect to μ if, for every vector measure γ X with values in B which is absolutely continuous with respect to μ, there is a μ-integrable function g : X → B such that

\gamma(E) = \int_E g\, d\mu

for every measurable set E. B has the Radon-Nikodym property if it has this property with respect to every finite measure. It is known that ℓ1 has the Radon-Nikodym property, but c0 does not.

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