Ba space

In mathematics, the ba space ${\displaystyle ba(\Sigma )}$ of an algebra of sets ${\displaystyle \Sigma }$ is the Banach space consisting of all bounded and finitely additive signed measures on ${\displaystyle \Sigma }$. The norm is defined as the variation, that is ${\displaystyle \|\nu \|=|\nu |(X).}$¹

If Σ is a sigma-algebra, then the space ${\displaystyle ca(\Sigma )}$ is defined as the subset of ${\displaystyle ba(\Sigma )}$ consisting of countably additive measures.² 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 ${\displaystyle rca(X)}$ is the subspace of ${\displaystyle ca(\Sigma )}$ 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 ${\displaystyle ca(\Sigma )}$ is a closed subset of ${\displaystyle ba(\Sigma )}$, and ${\displaystyle rca(X)}$ is a closed set of ${\displaystyle ca(\Sigma )}$ for Σ the algebra of Borel sets on X. The space of simple functions on ${\displaystyle \Sigma }$ is dense in ${\displaystyle ba(\Sigma )}$.

The ba space of the power set of the natural numbers, ba(2^N), is often denoted as simply ${\displaystyle ba}$ 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 Hildebrandt⁴ 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.

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 (${\displaystyle \mu (A)=\zeta \left(1_{A}\right)}$). 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:

${\displaystyle N_{\mu }:=\{f\in B(\Sigma ):f=0\ \mu {\text{-almost everywhere}}\}.}$

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

${\displaystyle N_{\mu }^{\perp }=\{\sigma \in ba(\Sigma ):\mu (A)=0\Rightarrow \sigma (A)=0{\text{ for any }}A\in \Sigma \},}$

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

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

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.