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 [math]\displaystyle{ (X, \Sigma, \mu) }[/math] be a measure space, and [math]\displaystyle{ B }[/math] be a Banach space. The Bochner integral of a function [math]\displaystyle{ f : X \to B }[/math] is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form [math]\displaystyle{ s(x) = \sum_{i=1}^n \chi_{E_i}(x) b_i }[/math] where the [math]\displaystyle{ E_i }[/math] are disjoint members of the [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \Sigma, }[/math] the [math]\displaystyle{ b_i }[/math] are distinct elements of [math]\displaystyle{ B, }[/math] and χ_E is the characteristic function of [math]\displaystyle{ E. }[/math] If [math]\displaystyle{ \mu\left(E_i\right) }[/math] is finite whenever [math]\displaystyle{ b_i \neq 0, }[/math] then the simple function is integrable, and the integral is then defined by [math]\displaystyle{ \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 }[/math] exactly as it is for the ordinary Lebesgue integral.

A measurable function [math]\displaystyle{ f : X \to B }[/math] is Bochner integrable if there exists a sequence of integrable simple functions [math]\displaystyle{ s_n }[/math] such that [math]\displaystyle{ \lim_{n\to\infty}\int_X \|f-s_n\|_B\,d\mu = 0, }[/math] where the integral on the left-hand side is an ordinary Lebesgue integral.

In this case, the Bochner integral is defined by [math]\displaystyle{ \int_X f\, d\mu = \lim_{n\to\infty}\int_X s_n\, d\mu. }[/math]

It can be shown that the sequence [math]\displaystyle{ \left\{\int_Xs_n\,d\mu \right\}_{n=1}^{\infty} }[/math] is a Cauchy sequence in the Banach space [math]\displaystyle{ B , }[/math] hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions [math]\displaystyle{ \{s_n\}_{n=1}^{\infty}. }[/math] These remarks show that the integral is well-defined (i.e independent of any choices). It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space [math]\displaystyle{ L^1. }[/math]

Properties

Elementary 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 [math]\displaystyle{ (X, \Sigma, \mu) }[/math] is a measure space, then a Bochner-measurable function [math]\displaystyle{ f \colon X \to B }[/math] is Bochner integrable if and only if [math]\displaystyle{ \int_X \|f\|_B\, \mathrm{d} \mu \lt \infty. }[/math]

Here, a function [math]\displaystyle{ f \colon X \to B }[/math] is called Bochner measurable if it is equal [math]\displaystyle{ \mu }[/math]-almost everywhere to a function [math]\displaystyle{ g }[/math] taking values in a separable subspace [math]\displaystyle{ B_0 }[/math] of [math]\displaystyle{ B }[/math], and such that the inverse image [math]\displaystyle{ g^{-1}(U) }[/math] of every open set [math]\displaystyle{ U }[/math] in [math]\displaystyle{ B }[/math] belongs to [math]\displaystyle{ \Sigma }[/math]. Equivalently, [math]\displaystyle{ f }[/math] is the limit [math]\displaystyle{ \mu }[/math]-almost everywhere of a sequence of countably-valued simple functions.

Linear operators

If [math]\displaystyle{ T \colon B \to B' }[/math] is a continuous linear operator between Banach spaces [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B' }[/math], and [math]\displaystyle{ f \colon X \to B }[/math] is Bochner integrable, then it is relatively straightforward to show that [math]\displaystyle{ T f \colon X \to B' }[/math] is Bochner integrable and integration and the application of [math]\displaystyle{ T }[/math] may be interchanged: [math]\displaystyle{ \int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu }[/math] for all measurable subsets [math]\displaystyle{ E \in \Sigma }[/math].

A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators. If [math]\displaystyle{ T \colon B \to B' }[/math] is a closed linear operator between Banach spaces [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B' }[/math] and both [math]\displaystyle{ f \colon X \to B }[/math] and [math]\displaystyle{ T f \colon X \to B' }[/math] are Bochner integrable, then [math]\displaystyle{ \int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu }[/math] for all measurable subsets [math]\displaystyle{ E \in \Sigma }[/math].

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if [math]\displaystyle{ f_n \colon X \to B }[/math] is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function [math]\displaystyle{ f }[/math], and if [math]\displaystyle{ \|f_n(x)\|_B \leq g(x) }[/math] for almost every [math]\displaystyle{ x \in X }[/math], and [math]\displaystyle{ g \in L^1(\mu) }[/math], then [math]\displaystyle{ \int_E \|f-f_n\|_B \, \mathrm{d} \mu \to 0 }[/math] as [math]\displaystyle{ n \to \infty }[/math] and [math]\displaystyle{ \int_E f_n\, \mathrm{d} \mu \to \int_E f \, \mathrm{d} \mu }[/math] for all [math]\displaystyle{ E \in \Sigma }[/math].

If [math]\displaystyle{ f }[/math] is Bochner integrable, then the inequality [math]\displaystyle{ \left\|\int_E f \, \mathrm{d} \mu\right\|_B \leq \int_E \|f\|_B \, \mathrm{d} \mu }[/math] holds for all [math]\displaystyle{ E \in \Sigma. }[/math] In particular, the set function [math]\displaystyle{ E\mapsto \int_E f\, \mathrm{d} \mu }[/math] defines a countably-additive [math]\displaystyle{ B }[/math]-valued vector measure on [math]\displaystyle{ X }[/math] which is absolutely continuous with respect to [math]\displaystyle{ \mu }[/math].

Radon–Nikodym property

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.

Specifically, if [math]\displaystyle{ \mu }[/math] is a measure on [math]\displaystyle{ (X, \Sigma), }[/math] then [math]\displaystyle{ B }[/math] has the Radon–Nikodym property with respect to [math]\displaystyle{ \mu }[/math] if, for every countably-additive vector measure [math]\displaystyle{ \gamma }[/math] on [math]\displaystyle{ (X, \Sigma) }[/math] with values in [math]\displaystyle{ B }[/math] which has bounded variation and is absolutely continuous with respect to [math]\displaystyle{ \mu, }[/math] there is a [math]\displaystyle{ \mu }[/math]-integrable function [math]\displaystyle{ g : X \to B }[/math] such that [math]\displaystyle{ \gamma(E) = \int_E g\, d\mu }[/math] for every measurable set [math]\displaystyle{ E \in \Sigma. }[/math]

The Banach space [math]\displaystyle{ B }[/math] has the Radon–Nikodym property if [math]\displaystyle{ B }[/math] has the Radon–Nikodym property with respect to every finite measure. Equivalent formulations include:

• Bounded discrete-time martingales in [math]\displaystyle{ B }[/math] converge a.s.
• Functions of bounded-variation into [math]\displaystyle{ B }[/math] are differentiable a.e.
• For every bounded [math]\displaystyle{ D\subseteq B }[/math], there exists [math]\displaystyle{ f\in B^* }[/math] and [math]\displaystyle{ \delta\in\mathbb{R}^+ }[/math] such that [math]\displaystyle{ \{x:f(x)+\delta\gt \sup{f(D)}\}\subseteq D }[/math] has arbitrarily small diameter.

It is known that the space [math]\displaystyle{ \ell 1 }[/math] has the Radon–Nikodym property, but [math]\displaystyle{ c 0 }[/math] and the spaces [math]\displaystyle{ L^{\infty}(\Omega), }[/math] [math]\displaystyle{ L^1(\Omega), }[/math] for [math]\displaystyle{ \Omega }[/math] an open bounded subset of [math]\displaystyle{ \R^n, }[/math] and [math]\displaystyle{ C(K), }[/math] for [math]\displaystyle{ K }[/math] 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 – Mathematical concept
• Bochner measurable function
• Pettis integral
• Vector measure
• Weakly measurable function

References

• Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind", Fundamenta Mathematicae 20: 262–276, http://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20127.pdf 
• Bourgin, Richard D. (1983). Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics 993. Berlin: Springer-Verlag. doi:10.1007/BFb0069321. ISBN 3-540-12296-6. 
• Cohn, Donald (2013), Measure Theory, Birkhäuser Advanced Texts Basler Lehrbücher, Springer, doi:10.1007/978-1-4614-6956-8, ISBN 978-1-4614-6955-1 
• Yosida, Kôsaku (1980), Functional Analysis, Classics in Mathematics, 123, Springer, doi:10.1007/978-3-642-61859-8, ISBN 978-3-540-58654-8 
• Diestel, Joseph (1984), Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, 92, Springer, doi:10.1007/978-1-4612-5200-9, ISBN 978-0-387-90859-5, https://archive.org/details/sequencesseriesi0000dies 
• Diestel; Uhl (1977), Vector measures, American Mathematical Society, ISBN 978-0-8218-1515-1, http://projecteuclid.org/euclid.bams/1183540941 
• Hille, Einar; Phillips, Ralph (1957), Functional Analysis and Semi-Groups, American Mathematical Society, ISBN 978-0-8218-1031-6 
• Lang, Serge (1993), Real and Functional Analysis (3rd ed.), Springer, ISBN 978-0387940014 
• Hazewinkel, Michiel, ed. (2001), "Bochner integral", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b016710 
• Hazewinkel, Michiel, ed. (2001), "Vector measures", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=V/v096490