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

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

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

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

exactly as it is for the ordinary Lebesgue integral.

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

$\int_X f\, d\mu = \lim_{n\to\infty}\int_X s_n\, d\mu.$

It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space $L^1$.

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

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

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 B₀ 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 $T$ is a continuous linear operator, and $f$ is Bochner-integrable, then $T f$ is Bochner-integrable and integration and $T$ may be interchanged:

$\int_X T f d\mu = T \int_X f d\mu.$

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

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

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

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 ƒ is Bochner integrable, then the inequality

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

holds 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.

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 $\gamma$ 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

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

for every measurable set E ∈ Σ.

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 $l_1$ has the Radon–Nikodym property, but $c_0$ and the spaces $L^{\infty}(\Omega)$, $L^{1}(\Omega)$, for $\Omega$ an open bounded subset of $\mathbb{R}^n$, and $C(K)$, 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

• Bochner space
• Pettis integral
• Vector measure