Rca space

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

If Σ is a sigma-algebra, then the space $ca\left(\Sigma\right)$ is defined as the subset of $ba\left(\Sigma\right)$ 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 $rca\left(X\right)$ is the subspace of $ca\left(\Sigma\right)$ 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 $ca\left(\Sigma\right)$ is a closed subset of $ba\left(\Sigma\right)$, and $rca\left(X\right)$ is a closed set of $ca\left(\Sigma\right)$ for Σ the algebra of Borel sets on X. The space of simple functions on $\Sigma$ is dense in $ba\left(\Sigma\right)$.

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 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.

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 ($\mu\left(A\right)=\zeta\left\left(1_A\right\right)$). 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.

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

$L^1\left(\mu\right)\subset L^1\left(\mu\right)^\left\{**\right\}=L^\left\{\infty\right\}\left(\mu\right)^*$

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.