ba space
In mathematics, the ba space of an algebra of sets is the
If Σ is a
If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then is the subspace of consisting of all regular Borel measures on X.[3]
Properties
All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus is a closed subset of , and is a closed set of for Σ the algebra of Borel sets on X. The space of simple functions on is dense in .
The ba space of the power set of the natural numbers, ba(2N), is often denoted as simply and is
Dual of B(Σ)
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the
The topological duality ba(Σ) = B(Σ)* is easy to see. There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(Σ)* if it is continuous in the sup-norm.
Dual of L∞(μ)
If Σ is a
The dual Banach space L∞(μ)* is thus isomorphic to
i.e. the space of
When the measure space is furthermore
is isomorphic to the inclusion of the space of countably additive μ-a.c. bounded measures inside the space of all finitely additive μ-a.c. bounded measures.
References
- Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience.
- ^ Dunford & Schwartz 1958, IV.2.15.
- ^ Dunford & Schwartz 1958, IV.2.16.
- ^ Dunford & Schwartz 1958, IV.2.17.
- JSTOR 1989829.
- .
- ^ Dunford & Schwartz 1958.
- ^ Diestel, J.; Uhl, J.J. (1977). Vector measures. Mathematical Surveys. Vol. 15. American Mathematical Society. Chapter I.
Further reading
- Diestel, Joseph (1984). Sequences and series in Banach spaces. Springer-Verlag. OCLC 9556781.
- Yosida, K.; Hewitt, E. (1952). "Finitely additive measures". Transactions of the American Mathematical Society. 72 (1): 46–66. JSTOR 1990654.
- Kantorovitch, Leonid V.; Akilov, Gleb P. (1982). Functional Analysis. Pergamon. ISBN 978-0-08-023036-8.