ba space

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

If Σ is a sigma-algebra, then the space ca(Σ) is defined as the subset of ba(Σ) consisting of countably additive measures. (Dunford & Schwartz 1958, IV.2.16)

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

[edit] Properties

All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus ca(Σ) is a closed subset of ba(Σ), and rca(X) is a closed set of ca(Σ) for Σ the algebra of Borel sets on X. The space of simple functions on Σ is dense in ba(Σ).

The ba space of the power set of the natural numbers, ba(2N), is often denoted as simply ba and is isomorphic to the dual space of the ℓ∞ space.

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.

[edit] References

  • Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5 .
  • Diestel, J. & Uhl, J.J. (1977), Vector measures, vol. 15, Mathematical Surveys, 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: 868-875 .
  • Fichtenholz, G & Kantorovich, L.V. (1934), “Sur les opérationes linéaires dans l'espace des fonctions bornées”, Studia Mathematica 5: 69-98 .
  • Yosida, K & Hewitt, E, “Finitely additive measures”, Transactions of the American Mathematical Society 72: 46-66, <http://links.jstor.org/sici?sici=0002-9947(195201)72%3A1%3C46%3AFAM%3E2.0.CO%3B2-G> .