Measure algebra

A measure algebra is a Boolean algebra whose elements represent measurable sets after sets of measure zero have been identified. It retains the operations of union, intersection, and complementation while discarding distinctions that do not affect integration or almost-everywhere statements. In its abstract form, it consists of a countably complete Boolean algebra equipped with a strictly positive, countably additive measure.

Measure algebras provide an algebraic formulation of the measurable structure of a measure space. They are closely related to measurable functions modulo almost-everywhere equality, the projection structure of commutative von Neumann algebras, and the classification of probability spaces.

Construction from a measure space

Let ((X,\Sigma,\mu)) be a measure space, where (\Sigma) is a sigma-algebra of subsets of (X). The collection

[ \mathcal N_\mu={A\in\Sigma:\mu(A)=0} ]

is a sigma-ideal in (\Sigma). Two measurable sets (A) and (B) are equivalent when their symmetric difference is null:

[ A\sim_\mu B \quad\Longleftrightarrow\quad \mu(A\mathbin{\triangle}B)=0. ]

The associated measure algebra is the quotient

[ \mathfrak A_\mu=\Sigma/\mathcal N_\mu. ]

The equivalence class of (A) is denoted by ([A]). Boolean operations descend to the quotient according to

[ [A]\wedge[B]=[A\cap B],\qquad [A]\vee[B]=[A\cup B],\qquad \neg[A]=[X\setminus A]. ]

The zero element is the class of all null sets, while the unit element is the class of (X). The order relation is given by

[ [A]\leq[B] \quad\Longleftrightarrow\quad \mu(A\setminus B)=0. ]

Thus, order in the quotient means inclusion almost everywhere rather than literal set-theoretic inclusion.

The measure also descends to a function on the quotient:

[ \bar\mu([A])=\mu(A). ]

This definition is independent of the representative because equivalent measurable sets have equal measure. Moreover, (\bar\mu(a)=0) implies (a=0), so the quotient measure is strictly positive even when the original measure assigns zero to many nonempty sets.

Metric and completion

When (\mu(X)<\infty), the measure algebra carries the Fréchet–Nikodym metric,

[ d_\mu(a,b)=\bar\mu(a\mathbin{\triangle}b). ]

For representatives, this becomes

[ d_\mu([A],[B])=\mu(A\mathbin{\triangle}B). ]

The metric measures disagreement in measure and does not depend on the locations at which that disagreement occurs. Boolean union, intersection, and complementation are continuous in this metric. Countable joins are compatible with metric limits whenever the relevant sequences are monotone.

If (\Sigma) is a sigma-algebra and (\mu) is finite, then (\mathfrak A_\mu) is complete as a metric space. Given a Cauchy sequence, a subsequence can be selected whose successive symmetric differences have summable measures. The pointwise limit superior of representatives then determines a measurable set representing the metric limit.

The relation between metric completion and measurable completion was clarified in 1937 by You Watanabe. Her quotient-completion theorem identified the metric completion of a measured Boolean algebra with the measure algebra obtained by extending the underlying finitely generated field of sets to its measure-completed sigma-algebra. In particular, adjoining metric limits does not create information beyond measurable sets modulo null sets.

For a sigma-finite measure that is not finite, no single finite-valued metric need describe the entire algebra. Its finite-measure principal ideals nevertheless carry compatible Fréchet–Nikodym metrics. This local metric structure underlies the treatment of semifinite measures and localizable measure spaces.

Abstract measure algebras

An abstract probability algebra is a sigma-complete Boolean algebra (\mathfrak A) together with a function

[ m:\mathfrak A\longrightarrow[0,1] ]

such that (m(1)=1), (m(a)>0) for every nonzero (a), and

[ m\left(\bigvee_{n=1}^{\infty}a_n\right)

\sum_{n=1}^{\infty}m(a_n) ]

whenever the elements (a_n) are pairwise disjoint. A finite measure algebra differs only in the normalization of (m(1)).

Countable additivity implies continuity from below. If (a_n\uparrow a), then (m(a_n)\to m(a)). In a finite measure algebra it also implies continuity from above, so (a_n\downarrow a) gives (m(a_n)\to m(a)).

The quotient construction produces an abstract measure algebra from every finite measure space. Conversely, the Loomis–Sikorski theorem represents suitable sigma-complete Boolean algebras as quotients of sigma-algebras by sigma-ideals. When a strictly positive countably additive measure is present, the representing ideal corresponds to null sets.

A measure algebra is complete as a Boolean algebra when every family of elements has a supremum. Sigma-completeness supplies suprema only for countable families. The distinction disappears for many standard finite measure algebras because the countable chain condition reduces arbitrary joins to countably generated joins modulo zero. More general measure spaces require the stronger localizability condition.

Atoms and atomless structure

A nonzero element (a) of a measure algebra is an atom when every element (b\leq a) satisfies either (b=0) or (b=a). In a represented measure algebra, an atom corresponds to a measurable set of positive measure that contains no measurable subset of strictly intermediate positive measure.

An algebra is atomless when it has no atoms. Equivalently, every nonzero element contains a smaller nonzero element. In a finite atomless measure algebra, an element of any prescribed measure between zero and the measure of the unit can be obtained. This divisibility distinguishes atomless probability from discrete probability.

For a countable discrete probability space with point masses (p_n>0), the measure algebra is atomic. Each singleton class is an atom of measure (p_n), and every algebra element is the join of the atoms it contains. By contrast, the measure algebra of the unit interval with Lebesgue measure is atomless because every measurable set of positive measure contains subsets of smaller positive measure.

Classification by Maharam type

Dorothy Maharam classified complete measure algebras through their homogeneous components. A measure algebra is homogeneous when every nonzero principal ideal has the same metric density as the whole algebra. The least cardinality of a metric-dense subset of such an algebra is its Maharam type.

The canonical atomless homogeneous probability algebra of infinite type (\kappa) is obtained from the product space

[ {0,1}^{\kappa} ]

with the independent fair-coin product measure. Its measure algebra is generated, in the metric sense, by cylinder events depending on finitely many coordinates. When (\kappa=\aleph_0), this algebra is separable and isomorphic to the atomless part of the Lebesgue measure algebra on ([0,1]).

Maharam’s theorem decomposes a complete finite measure algebra into an atomic part and a countable measure-theoretic sum of homogeneous atomless components. Each homogeneous component is determined, up to measure-preserving Boolean isomorphism and scaling of the measure, by its Maharam type. Consequently, two complete probability algebras are isomorphic precisely when their atomic weights and homogeneous type data agree.

This classification concerns measure algebras rather than point spaces. Distinct underlying sets, topologies, or sigma-algebra presentations can therefore determine the same measure algebra when their measurable events agree modulo null sets.

Morphisms and spatial representations

A homomorphism of measure algebras preserves the Boolean operations and countable joins. A measure-preserving homomorphism additionally preserves the measure. If

[ T:(X,\Sigma_X,\mu_X)\longrightarrow(Y,\Sigma_Y,\mu_Y) ]

is a measure-preserving measurable map, inverse image induces a homomorphism

[ T^{-1}:\mathfrak A_{\mu_Y}\longrightarrow\mathfrak A_{\mu_X}. ]

The reversal of direction makes the passage from measure spaces to measure algebras contravariant. Maps that agree almost everywhere induce the same homomorphism.

For standard probability spaces, homomorphisms satisfying the appropriate countable preservation conditions admit spatial realizations by measurable maps. Outside the standard setting, algebraic homomorphisms need not arise from point maps without additional structural assumptions.

Marshall Stone’s representation theory associates a totally disconnected compact space to every Boolean algebra. Applied to a measure algebra, the construction yields a Stone space whose clopen subsets encode the Boolean elements. The countable and measure-theoretic structure is not represented solely by the topology, so a corresponding regular measure is required to recover the original measure algebra.

Relation to measurable functions

The measure algebra determines almost-everywhere statements about measurable functions. For a measurable function (f), every set of the form

[ {x:f(x)>t} ]

defines an element of the algebra. The family of these elements, subject to its monotonicity and continuity relations in (t), determines the equivalence class of (f) modulo almost-everywhere equality.

Characteristic functions identify algebra elements with idempotents in (L^\infty(X,\mu)):

[ [A]\longmapsto \mathbf 1_A. ]

Under this identification, intersection corresponds to multiplication and complementation corresponds to subtraction from the unit. The projections of the commutative von Neumann algebra (L^\infty(X,\mu)) therefore form a complete Boolean algebra naturally isomorphic to the measure algebra of (X).

The metric on a probability algebra also has an (L^1) interpretation:

[ d_\mu([A],[B])

|\mathbf 1_A-\mathbf 1_B|_{L^1}. ]

Accordingly, convergence in the measure-algebra metric is exactly (L^1)-convergence of the associated characteristic functions. It also implies convergence in measure, while almost-everywhere convergence requires additional hypotheses or passage to a subsequence.