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 ${\displaystyle (X,\Sigma ,\mu )}$  be a measure space, and ${\displaystyle B}$  be a Banach space. The Bochner integral of a function ${\displaystyle f:X\to B}$  is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form

$${\displaystyle s(x)=\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i}}$$

 

where the ${\displaystyle E_{i}}$  are disjoint members of the ${\displaystyle \sigma }$ -algebra ${\displaystyle \Sigma ,}$  the ${\displaystyle b_{i}}$  are distinct elements of ${\displaystyle B,}$  and χ_E is the characteristic function of ${\displaystyle E.}$  If ${\displaystyle \mu \left(E_{i}\right)}$  is finite whenever ${\displaystyle b_{i}\neq 0,}$  then the simple function is integrable, and the integral is then defined by

$${\displaystyle \int _{X}\left[\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i}\right]\,d\mu =\sum _{i=1}^{n}\mu (E_{i})b_{i}}$$

 

exactly as it is for the ordinary Lebesgue integral.

A measurable function ${\displaystyle f:X\to B}$  is Bochner integrable if there exists a sequence of integrable simple functions ${\displaystyle s_{n}}$  such that

$${\displaystyle \lim _{n\to \infty }\int _{X}\|f-s_{n}\|_{B}\,d\mu =0,}$$

 

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

In this case, the Bochner integral is defined by

$${\displaystyle \int _{X}f\,d\mu =\lim _{n\to \infty }\int _{X}s_{n}\,d\mu .}$$

 

It can be shown that the sequence ${\displaystyle \left\{\int _{X}s_{n}\,d\mu \right\}_{n=1}^{\infty }}$  is a Cauchy sequence in the Banach space ${\displaystyle B,}$  hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions ${\displaystyle \{s_{n}\}_{n=1}^{\infty }.}$  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 ${\displaystyle L^{1}.}$ 

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

$${\displaystyle \int _{X}\|f\|_{B}\,\mathrm {d} \mu <\infty .}$$

 

Here, a function ${\displaystyle f\colon X\to B}$  is called Bochner measurable if it is equal ${\displaystyle \mu }$ -almost everywhere to a function ${\displaystyle g}$  taking values in a separable subspace ${\displaystyle B_{0}}$  of ${\displaystyle B}$ , and such that the inverse image ${\displaystyle g^{-1}(U)}$  of every open set ${\displaystyle U}$  in ${\displaystyle B}$  belongs to ${\displaystyle \Sigma }$ . Equivalently, ${\displaystyle f}$  is the limit ${\displaystyle \mu }$ -almost everywhere of a sequence of countably-valued simple functions.

Linear operators

If ${\displaystyle T\colon B\to B'}$  is a continuous linear operator between Banach spaces ${\displaystyle B}$  and ${\displaystyle B'}$ , and ${\displaystyle f\colon X\to B}$  is Bochner integrable, then it is relatively straightforward to show that ${\displaystyle Tf\colon X\to B'}$  is Bochner integrable and integration and the application of ${\displaystyle T}$  may be interchanged:

$${\displaystyle \int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu }$$

 

for all measurable subsets ${\displaystyle E\in \Sigma }$ .

A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.^[1] If ${\displaystyle T\colon B\to B'}$  is a closed linear operator between Banach spaces ${\displaystyle B}$  and ${\displaystyle B'}$  and both ${\displaystyle f\colon X\to B}$  and ${\displaystyle Tf\colon X\to B'}$  are Bochner integrable, then

$${\displaystyle \int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu }$$

 

for all measurable subsets ${\displaystyle E\in \Sigma }$ .

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if ${\displaystyle f_{n}\colon X\to B}$  is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function ${\displaystyle f}$ , and if

$${\displaystyle \|f_{n}(x)\|_{B}\leq g(x)}$$

 

for almost every ${\displaystyle x\in X}$ , and ${\displaystyle g\in L^{1}(\mu )}$ , then

$${\displaystyle \int _{E}\|f-f_{n}\|_{B}\,\mathrm {d} \mu \to 0}$$

 

as ${\displaystyle n\to \infty }$  and

$${\displaystyle \int _{E}f_{n}\,\mathrm {d} \mu \to \int _{E}f\,\mathrm {d} \mu }$$

 

for all ${\displaystyle E\in \Sigma }$ .

If ${\displaystyle f}$  is Bochner integrable, then the inequality

$${\displaystyle \left\|\int _{E}f\,\mathrm {d} \mu \right\|_{B}\leq \int _{E}\|f\|_{B}\,\mathrm {d} \mu }$$

 

holds for all ${\displaystyle E\in \Sigma .}$  In particular, the set function

$${\displaystyle E\mapsto \int _{E}f\,\mathrm {d} \mu }$$

 

defines a countably-additive ${\displaystyle B}$ -valued vector measure on ${\displaystyle X}$  which is absolutely continuous with respect to ${\displaystyle \mu }$ .

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 ${\displaystyle \mu }$  is a measure on ${\displaystyle (X,\Sigma ),}$  then ${\displaystyle B}$  has the Radon–Nikodym property with respect to ${\displaystyle \mu }$  if, for every countably-additive vector measure ${\displaystyle \gamma }$  on ${\displaystyle (X,\Sigma )}$  with values in ${\displaystyle B}$  which has bounded variation and is absolutely continuous with respect to ${\displaystyle \mu ,}$  there is a ${\displaystyle \mu }$ -integrable function ${\displaystyle g:X\to B}$  such that

$${\displaystyle \gamma (E)=\int _{E}g\,d\mu }$$

 

for every measurable set ${\displaystyle E\in \Sigma .}$ ^[2]

The Banach space ${\displaystyle B}$  has the Radon–Nikodym property if ${\displaystyle B}$  has the Radon–Nikodym property with respect to every finite measure.^[2] Equivalent formulations include:

• Bounded discrete-time martingales in ${\displaystyle B}$  converge a.s.^[3]
• Functions of bounded-variation into ${\displaystyle B}$  are differentiable a.e.^[4]
• For every bounded ${\displaystyle D\subseteq B}$ , there exists ${\displaystyle f\in B^{*}}$  and ${\displaystyle \delta \in \mathbb {R} ^{+}}$  such that
  $${\displaystyle \{x:f(x)+\delta >\sup {f(D)}\}\subseteq D}$$

   
  has arbitrarily small diameter.^[3]

It is known that the space ${\displaystyle \ell _{1}}$  has the Radon–Nikodym property, but ${\displaystyle c_{0}}$  and the spaces ${\displaystyle L^{\infty }(\Omega ),}$  ${\displaystyle L^{1}(\Omega ),}$  for ${\displaystyle \Omega }$  an open bounded subset of ${\displaystyle \mathbb {R} ^{n},}$  and ${\displaystyle C(K),}$  for ${\displaystyle K}$  an infinite compact space, do not.^[5] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem)^{[citation needed]} and reflexive spaces, which include, in particular, Hilbert spaces.^[2]

See also

• Bochner space – Type of topological space
• Bochner measurable function
• Pettis integral
• Vector measure
• Weakly measurable function

References

1. Diestel, Joseph; Uhl, Jr., John Jerry (1977). Vector Measures. Mathematical Surveys. American Mathematical Society. doi:10.1090/surv/015. (See Theorem II.2.6)
2. Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56].
3. Bourgin 1983, pp. 31, 33. Thm. 2.3.6-7, conditions (1,4,10).
4. Bourgin 1983, p. 16. "Early workers in this field were concerned with the Banach space property that each X-valued function of bounded variation on [0,1] 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. Bourgin 1983, p. 14.

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• 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.
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