Английская Википедия: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 <math>(X, \Sigma, \mu)</math> be a measure space, and <math>B</math> be a Banach space. The Bochner integral of a function <math>f : X \to B</math> is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form <math display="block">s(x) = \sum_{i=1}^n \chi_{E_i}(x) b_i</math> where the <math>E_i</math> are disjoint members of the <math>\sigma</math>-algebra <math>\Sigma,</math> the <math>b_i</math> are distinct elements of <math>B,</math> and χE is the characteristic function of <math>E.</math> If <math>\mu\left(E_i\right)</math> is finite whenever <math>b_i \neq 0,</math> then the simple function is integrable, and the integral is then defined by <math display="block">\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</math> exactly as it is for the ordinary Lebesgue integral.
A measurable function <math>f : X \to B</math> is Bochner integrable if there exists a sequence of integrable simple functions <math>s_n</math> such that <math display="block">\lim_{n\to\infty}\int_X \|f-s_n\|_B\,d\mu = 0,</math> where the integral on the left-hand side is an ordinary Lebesgue integral.
In this case, the Bochner integral is defined by <math display="block">\int_X f\, d\mu = \lim_{n\to\infty}\int_X s_n\, d\mu.</math>
It can be shown that the sequence <math> \left\{\int_Xs_n\,d\mu \right\}_{n=1}^{\infty} </math> is a Cauchy sequence in the Banach space <math> B ,</math> hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions <math>\{s_n\}_{n=1}^{\infty}.</math> 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 <math>L^1.</math>
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 <math>(X, \Sigma, \mu)</math> is a measure space, then a Bochner-measurable function <math>f \colon X \to B</math> is Bochner integrable if and only if <math display="block">\int_X \|f\|_B\, \mathrm{d} \mu < \infty.</math>
Here, a function <math>f \colon X \to B</math> is called Bochner measurable if it is equal <math>\mu</math>-almost everywhere to a function <math>g</math> taking values in a separable subspace <math>B_0</math> of <math>B</math>, and such that the inverse image <math>g^{-1}(U)</math> of every open set <math>U</math> in <math>B</math> belongs to <math>\Sigma</math>. Equivalently, <math>f</math> is the limit <math>\mu</math>-almost everywhere of a sequence of countably-valued simple functions.
Linear operators
If <math>T \colon B \to B'</math> is a continuous linear operator between Banach spaces <math>B</math> and <math>B'</math>, and <math>f \colon X \to B</math> is Bochner integrable, then it is relatively straightforward to show that <math>T f \colon X \to B'</math> is Bochner integrable and integration and the application of <math>T</math> may be interchanged: <math display="block">\int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu</math> for all measurable subsets <math>E \in \Sigma</math>.
A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.[1] If <math>T \colon B \to B'</math> is a closed linear operator between Banach spaces <math>B</math> and <math>B'</math> and both <math>f \colon X \to B</math> and <math>T f \colon X \to B'</math> are Bochner integrable, then <math display="block">\int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu</math> for all measurable subsets <math>E \in \Sigma</math>.
Dominated convergence theorem
A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if <math>f_n \colon X \to B</math> is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function <math>f</math>, and if <math display="block">\|f_n(x)\|_B \leq g(x)</math> for almost every <math>x \in X</math>, and <math>g \in L^1(\mu)</math>, then <math display="block">\int_E \|f-f_n\|_B \, \mathrm{d} \mu \to 0</math> as <math>n \to \infty</math> and <math display="block">\int_E f_n\, \mathrm{d} \mu \to \int_E f \, \mathrm{d} \mu</math> for all <math>E \in \Sigma</math>.
If <math>f</math> is Bochner integrable, then the inequality <math display="block">\left\|\int_E f \, \mathrm{d} \mu\right\|_B \leq \int_E \|f\|_B \, \mathrm{d} \mu</math> holds for all <math>E \in \Sigma.</math> In particular, the set function <math display="block">E\mapsto \int_E f\, \mathrm{d} \mu</math> defines a countably-additive <math>B</math>-valued vector measure on <math>X</math> which is absolutely continuous with respect to <math>\mu</math>.
Radon–Nikodym property
An important fact about the Bochner integral is that the Radon–Nikodym theorem Шаблон:Em to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of nice Banach spaces.
Specifically, if <math>\mu</math> is a measure on <math>(X, \Sigma),</math> then <math>B</math> has the Radon–Nikodym property with respect to <math>\mu</math> if, for every countably-additive vector measure <math>\gamma</math> on <math>(X, \Sigma)</math> with values in <math>B</math> which has bounded variation and is absolutely continuous with respect to <math>\mu,</math> there is a <math>\mu</math>-integrable function <math>g : X \to B</math> such that <math display="block">\gamma(E) = \int_E g\, d\mu </math> for every measurable set <math>E \in \Sigma.</math>[2]
The Banach space <math>B</math> has the Radon–Nikodym property if <math>B</math> has the Radon–Nikodym property with respect to every finite measure.[2] Equivalent formulations include:
- Bounded discrete-time martingales in <math>B</math> converge a.s.[3]
- Functions of bounded-variation into <math>B</math> are differentiable a.e.[4]
- For every bounded <math>D\subseteq B</math>, there exists <math>f\in B^*</math> and <math>\delta\in\mathbb{R}^+</math> such that <math display=block>\{x:f(x)+\delta>\sup{f(D)}\}\subseteq D</math> has arbitrarily small diameter.[3]
It is known that the space <math>\ell_1</math> has the Radon–Nikodym property, but <math>c_0</math> and the spaces <math>L^{\infty}(\Omega),</math> <math>L^1(\Omega),</math> for <math>\Omega</math> an open bounded subset of <math>\R^n,</math> and <math>C(K),</math> for <math>K</math> an infinite compact space, do not.[5] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem)Шаблон:Cite needed and reflexive spaces, which include, in particular, Hilbert spaces.[2]
See also
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
References
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Springer
- Шаблон:Springer
Шаблон:Integral Шаблон:Functional analysis Шаблон:Analysis in topological vector spaces
- ↑ Шаблон:Cite book (See Theorem II.2.6)
- ↑ 2,0 2,1 2,2 Шаблон:Cite journal
- ↑ 3,0 3,1 Шаблон:Harvnb. Thm. 2.3.6-7, conditions (1,4,10).
- ↑ Шаблон:Harvnb. "Early workers in this field were concerned with the Banach space property that each Шаблон:Mvar-valued function of bounded variation on Шаблон:Closed-closed 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]."
- ↑ Шаблон:Harvnb.
