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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