# 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 χE 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 $\displaystyle{ \left\{\int_Xs_n\,d\mu \right\}_{n=1}^{\infty} }$ 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 $\displaystyle{ \int_X \|f\|_B\, \mathrm{d} \mu \lt \infty. }$

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 $\displaystyle{ T f \colon X \to B' }$ is Bochner integrable and integration and the application of $\displaystyle{ T }$ may be interchanged: $\displaystyle{ \int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu }$ 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 $\displaystyle{ T f \colon X \to B' }$ are Bochner integrable, then $\displaystyle{ \int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu }$ 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 $\displaystyle{ \mu }$.

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 $\displaystyle{ \{x:f(x)+\delta\gt \sup{f(D)}\}\subseteq D }$ 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 $\displaystyle{ \R^n, }$ 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]