Misplaced Pages

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 ${\displaystyle \|\nu \|=|\nu |(X).}$

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. 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.

Properties

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), is often denoted as simply ${\displaystyle ba}$ 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 and Fichtenholtz & Kantorovich. 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, and is often used to define the integral with respect to vector measures, 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.

Dual of 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:

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

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 L(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.

See also

• List of Banach spaces

References

• Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience.

1. Dunford & Schwartz 1958, IV.2.15.
2. Dunford & Schwartz 1958, IV.2.16.
3. Dunford & Schwartz 1958, IV.2.17.
4. Hildebrandt, T.H. (1934). "On bounded functional operations". Transactions of the American Mathematical Society. 36 (4): 868–875. doi:10.2307/1989829. JSTOR 1989829.
5. 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.
6. Dunford & Schwartz 1958.
7. Diestel, J.; Uhl, J.J. (1977). Vector measures. Mathematical Surveys. Vol. 15. American Mathematical Society. Chapter I.

Further reading

• Diestel, Joseph (1984). Sequences and series in Banach spaces. Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
• Yosida, K.; Hewitt, E. (1952). "Finitely additive measures". Transactions of the American Mathematical Society. 72 (1): 46–66. doi:10.2307/1990654. JSTOR 1990654.
• Kantorovitch, Leonid V.; Akilov, Gleb P. (1982). Functional Analysis. Pergamon. doi:10.1016/C2013-0-03044-7. ISBN 978-0-08-023036-8.

• Measure theory
• Banach spaces