Ba Space

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## Properties

### Dual of B(?)

### Dual of *L*^{?}(*?*)

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Ba Space

In mathematics, the **ba space** of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is (Dunford & Schwartz 1958, IV.2.15)

If ? is a sigma-algebra, then the space is defined as the subset of consisting of countably additive measures. (Dunford & Schwartz 1958, IV.2.16) The notation *ba* is a mnemonic for *bounded additive* and *ca* is short for *countably additive*.

If *X* is a topological space, and ? is the sigma-algebra of Borel sets in *X*, then is the subspace of consisting of all regular Borel measures on *X*. (Dunford & Schwartz 1958, IV.2.17)

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 ? 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*(2^{N}), is often denoted as simply and is isomorphic to the dual space of the l^{?} space.

Let B(?) be the space of bounded ?-measurable functions, equipped with the uniform norm. Then *ba*(?) = B(?)* is the continuous dual space of B(?). This is due to Hildebrandt (1934) and Fichtenholtz & Kantorovich (1934) . This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to *define* the integral with respect to a finitely additive measure (note that the usual Lebesgue integral requires *countable* additivity). This is due to Dunford & Schwartz (1958), and is often used to define the integral with respect to vector measures (Diestel & Uhl 1977, Chapter I), and especially vector-valued Radon measures.

The topological duality *ba*(?) = B(?)* is easy to see. There is an obvious *algebraic* duality between the vector space of *all* finitely additive measures ? on ? and the vector space of simple functions (). It is easy to check that the linear form induced by ? is continuous in the sup-norm iff ? is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(?)* iff it is continuous in the sup-norm.

If ? is a sigma-algebra and *?* is a sigma-additive positive measure on ? then the Lp space *L*^{?}(*?*) endowed with the essential supremum norm is by definition the quotient space of B(?) by the closed subspace of bounded *?*-null functions:

The dual Banach space *L*^{?}(*?*)* is thus isomorphic to

i.e. the space of finitely additive signed measures on *?* that are absolutely continuous with respect to *?* (*?*-a.c. for short).

When the measure space is furthermore sigma-finite then *L*^{?}(*?*) is in turn dual to *L*^{1}(*?*), which by the Radon-Nikodym theorem is identified with the set of all countably additive *?*-a.c. measures.
In other words, the inclusion in the bidual

is isomorphic to the inclusion of the space of countably additive *?*-a.c. bounded measures inside the space of all finitely additive *?*-a.c. bounded measures.

- Diestel, Joseph (1984),
*Sequences and series in Banach spaces*, Springer-Verlag, ISBN 0-387-90859-5, OCLC 9556781. - Diestel, J.; Uhl, J.J. (1977),
*Vector measures*, Mathematical Surveys,**15**, American Mathematical Society. - Dunford, N.; Schwartz, J.T. (1958),
*Linear operators, Part I*, Wiley-Interscience. - Hildebrandt, T.H. (1934), "On bounded functional operations",
*Transactions of the American Mathematical Society*,**36**(4): 868-875, doi:10.2307/1989829, JSTOR 1989829. - Fichtenholz, G; Kantorovich, L.V. (1934), "Sur les opérations linéaires dans l'espace des fonctions bornées",
*Studia Mathematica*,**5**: 69-98, doi:10.4064/sm-5-1-69-98. - Yosida, K; Hewitt, E (1952), "Finitely additive measures",
*Transactions of the American Mathematical Society*,**72**(1): 46-66, doi:10.2307/1990654, JSTOR 1990654.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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