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 be a measure space, and be a Banach space. The Bochner integral of a function is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form
A measurable function is Bochner integrable if there exists a sequence of integrable simple functions such that
In this case, the Bochner integral is defined by
It can be shown that the sequence is a Cauchy sequence in the Banach space hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions 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
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 is a measure space, then a Bochner-measurable function is Bochner integrable if and only if
A function is called Bochner-measurable if it is equal -almost everywhere to a function taking values in a separable subspace of and such that the inverse image of every open set in belongs to Equivalently, is limit -almost everywhere of a sequence of simple functions.
If is a continuous linear operator, and is Bochner-integrable, then is Bochner-integrable and integration and may be interchanged:
This also holds for closed operators, given that be itself integrable (which, via the criterion mentioned above is trivially true for bounded ).
A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function and if
If is Bochner integrable, then the inequality
Radon–Nikodym property[]
An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general. This results in an important property of Banach spaces known as the Radon–Nikodym property. Specifically, if is a measure on then has the Radon–Nikodym property with respect to if, for every countably-additive vector measure on with values in which has bounded variation and is absolutely continuous with respect to there is a -integrable function such that
The Banach space has the Radon–Nikodym property if has the Radon–Nikodym property with respect to every finite measure. It is known that the space has the Radon–Nikodym property, but and the spaces for an open bounded subset of and for an infinite compact space, do not.[citation needed] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem) and reflexive spaces, which include, in particular, Hilbert spaces.
See also[]
- Bochner space – Mathematical concept
- Bochner measurable function
- Pettis integral
- Vector measure
- Weakly measurable function
References[]
- ^ Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56].
- Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" (PDF), Fundamenta Mathematicae, 20: 262–276
- 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
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- 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
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- 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
- Definitions of mathematical integration
- Topological vector spaces
- Integral representations