# 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 Σ 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 Σ 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)

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

### Dual of B(Σ)

Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm. Then *ba*(Σ) = B(Σ)* is the continuous dual space of B(Σ). 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*(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 iff 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)* iff it is continuous in the sup-norm.

### Dual of *L*^{∞}(*μ*)

*L*

^{∞}(

*μ*)

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

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

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

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

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

- 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.