Bochner integral explained

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, \Sigma, \mu) be a measure space, and B be a Banach space. The Bochner integral of a function f : X \to B is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form$$s(x)\; =\; \backslash sum\_^n\; \backslash chi\_(x)\; b\_i,$$where the E _{i} are disjoint members of the \sigma -algebra \Sigma, the b _{i} are distinct elements of B, and χ_E is the characteristic function of E. If \mu\left(E _{i\right)} is finite whenever b _{i} ≠ 0 , then the simple function is integrable, and the integral is then defined by$$\backslash int\_X\; \backslash left[\backslash sum\_\{i=1\}^n\; \backslash chi\_\{E\_i\}(x)\; b\_i\backslash right]\backslash ,\; d\backslash mu\; =\; \backslash sum\_^n\; \backslash mu(E\_i)\; b\_i$$exactly as it is for the ordinary Lebesgue integral.

A measurable function

f : X \to B is Bochner integrable if there exists a sequence of integrable simple functions s _{n} such that$$\backslash lim\_\backslash int\_X\; \backslash |f-s\_n\backslash |\_B\backslash ,d\backslash mu\; =\; 0,$$where the integral on the left-hand side is an ordinary Lebesgue integral.

In this case, the Bochner integral is defined by$$\backslash int\_X\; f\backslash ,\; d\backslash mu\; =\; \backslash lim\_\backslash int\_X\; s\_n\backslash ,\; d\backslash mu.$$

It can be shown that the sequence

\left\{\int _{Xs} _{nd\mu} \right\} ^{inf ty} _{n =1} is a Cauchy sequence in the Banach space B , hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions \{s _{n\}} ^{inf ty} ^{n =1} . 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 L ^{1.}

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

(X, \Sigma, \mu) is a measure space, then a Bochner-measurable function f \colon X \to B is Bochner integrable if and only if

Here, a function

f \colon X \to B  is called Bochner measurable if it is equal \mu -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 \Sigma . Equivalently, f is the limit \mu -almost everywhere of a sequence of countably-valued simple functions.

Linear operators

If

T \colon B \to B' is a continuous linear operator between Banach spaces B and B' , and f \colon X \to B is Bochner integrable, then it is relatively straightforward to show that T f \colon X \to B' is Bochner integrable and integration and the application of T may be interchanged:$$\backslash int\_E\; T\; f\; \backslash ,\; \backslash mathrm\; \backslash mu\; =\; T\; \backslash int\_E\; f\; \backslash ,\; \backslash mathrm\; \backslash mu$$for all measurable subsets E \in \Sigma .

A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.^[1] If

T \colon B \to B' is a closed linear operator between Banach spaces B and B' and both f \colon X \to B and T f \colon X \to B' are Bochner integrable, then$$\backslash int\_E\; T\; f\; \backslash ,\; \backslash mathrm\; \backslash mu\; =\; T\; \backslash int\_E\; f\; \backslash ,\; \backslash mathrm\; \backslash mu$$for all measurable subsets E \in \Sigma .

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if

f _{n} \colon X \to B is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function f , and if$$\backslash |f\_n(x)\backslash |\_B\; \backslash leq\; g(x)$$for almost every x \in X , and g \in L ^{1(\mu)} , then$$\backslash int\_E\; \backslash |f-f\_n\backslash |\_B\; \backslash ,\; \backslash mathrm\; \backslash mu\; \backslash to\; 0$$as n \to inf ty and$$\backslash int\_E\; f\_n\backslash ,\; \backslash mathrm\; \backslash mu\; \backslash to\; \backslash int\_E\; f\; \backslash ,\; \backslash mathrm\; \backslash mu$$for all E \in \Sigma .

If

f is Bochner integrable, then the inequality$$\backslash left\backslash |\backslash int\_E\; f\; \backslash ,\; \backslash mathrm\; \backslash mu\backslash right\backslash |\_B\; \backslash leq\; \backslash int\_E\; \backslash |f\backslash |\_B\; \backslash ,\; \backslash mathrm\; \backslash mu$$holds for all E \in \Sigma. In particular, the set function$$E\backslash mapsto\; \backslash int\_E\; f\backslash ,\; \backslash mathrm\; \backslash mu$$defines a countably-additive B -valued vector measure on X which is absolutely continuous with respect to \mu .

Radon–Nikodym property

An important fact about the Bochner integral is that the Radon–Nikodym theorem to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of nice Banach spaces.

Specifically, if

\mu is a measure on (X, \Sigma), then B has the Radon–Nikodym property with respect to \mu if, for every countably-additive vector measure \gamma on (X, \Sigma) with values in B which has bounded variation and is absolutely continuous with respect to \mu, there is a \mu -integrable function g : X \to B such that$$\backslash gamma(E)\; =\; \backslash int\_E\; g\backslash ,\; d\backslash mu$$for every measurable set E \in \Sigma.

The Banach space

B has the Radon–Nikodym property if B has the Radon–Nikodym property with respect to every finite measure.^[2] Equivalent formulations include:

• Bounded discrete-time martingales in

B converge a.s.^[3]

• Functions of bounded-variation into

B are differentiable a.e.^[4]

• For every bounded

D\subseteq B , there exists f\in B ^{*} and \delta\in R ^{+} such that $$\backslash \backslash subseteq\; D$$ has arbitrarily small diameter.

It is known that the space

\ell _{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 \R ^{n,} and C(K), for K an infinite compact space, do not.^[5] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem)August 2022. and reflexive spaces, which include, in particular, Hilbert spaces.

References

• • Book: Bourgin, Richard D.. Lecture Notes in Mathematics 993. Springer-Verlag. Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Berlin. 1983. 10.1007/BFb0069321. 3-540-12296-6.

Notes and References

1. Book: Diestel. Joseph. Uhl, Jr.. John Jerry. Vector Measures. American Mathematical Society. Mathematical Surveys. 15. 1977. 10.1090/surv/015. (See Theorem II.2.6)
2. The Radon–Nikodym Theorem for Reflexive Banach Spaces. Diómedes. Bárcenas. Divulgaciones Matemáticas. 11. 1. 2003. 55–59 [pp. 55–56].
3. . Thm. 2.3.6-7, conditions (1,4,10).
4. . "Early workers in this field were concerned with the Banach space property that each -valued function of bounded variation on be differentiable almost surely. It turns out that this property (known as the Gelfand-Fréchet property) is also equivalent to the RNP [Radon-Nikodym Property]."
5. .