Ba space
Encyclopedia
In mathematics
, the ba space
of an algebra of sets
is the Banach space
consisting of all bounded and finitely additive measure
s on
. The norm is defined as the variation, that is 
If Σ is a sigma-algebra
, then the space
is defined as the subset of 
consisting of countably additive measures.
If X is a topological space, and Σ is the sigma-algebra of Borel set
s in X, then
is the subspace of 
consisting of all regular
Borel measures on X.
s) with respect to the same norm defined by the total variation, and thus
is a closed subset of 
, and 
is a closed set of 
for Σ the algebra of Borel sets on X. The space of simple function
s on
is dense in 
.
The ba space of the power set of the natural number
s, ba(2N), is often denoted as simply
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 and . 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 , and is often used to define the integral with respect to vector measure
s , and especially vector-valued Radon measure
s.
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 function
s. 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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the ba space
of an algebra of setsAlgebra of sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...
is the Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
consisting of all bounded and finitely additive measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
s on
. The norm is defined as the variation, that is If Σ is a sigma-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
, then the space
is defined as the subset of consisting of countably additive measures. If X is a topological space, and Σ is the sigma-algebra of Borel set
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
s in X, then
is the subspace of consisting of all regularRegular measure
In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...
Borel measures on X.
Properties
All three spaces are complete (they are Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s) with respect to the same norm defined by the total variation, and thus
is a closed subset of , and is a closed set of for Σ the algebra of Borel sets on X. The space of simple functionSimple function
In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains only a finite number of values...
s on
is dense in .The ba space of the power set of the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, ba(2N), is often denoted as simply
and is isomorphic to the dual spaceDual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of the ℓ∞ space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
.
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 and . This is a kind of Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space 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
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
with respect to a finitely additive measure (note that the usual Lebesgue integral requires countable additivity). This is due to , and is often used to define the integral with respect to vector measure
Vector measure
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of measure, which takes nonnegative real values only.-Definitions and first consequences:...
s , and especially vector-valued Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
s.
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 function
Simple function
In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains only a finite number of values...
s. 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
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
and μ is a sigma-additive positive measure on Σ then the Lp space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
L∞(μ) endowed with the essential supremum norm is by definition the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
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.