Bochner integral

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 ${\displaystyle L^{1}.}$

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) and reflexive spaces, which include, in particular, Hilbert spaces.[2]

• Bochner space – Mathematical concept
• Bochner measurable function
• Pettis integral
• Vector measure
• Weakly measurable function

• Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" (PDF), Fundamenta Mathematicae, 20: 262–276
• 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, vol. 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, vol. 92, Springer, doi:10.1007/978-1-4612-5200-9, ISBN 978-0-387-90859-5
• Diestel; Uhl (1977), Vector measures, American Mathematical Society, ISBN 978-0-8218-1515-1
• 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
• Sobolev, V. I. (2001) [1994], "Bochner integral", Encyclopedia of Mathematics, EMS Press
• van Dulst, D. (2001) [1994], "Vector measures", Encyclopedia of Mathematics, EMS Press