is the Banach space consisting of all bounded and finitely additive signed measures on
consisting of countably additive measures.
If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then
consisting of all regular Borel measures on X.
[3] All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus
for Σ the algebra of Borel sets on X.
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm.
This is due to Hildebrandt[4] and Fichtenholtz & Kantorovich.
[5] 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,[6] and is often used to define the integral with respect to vector measures,[7] and especially vector-valued Radon measures.
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.
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 L1(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.