nLab Maharam's theorem

Idea

Maharam’s theorem states a complete classification of isomorphism classes of the appropriate category of measurable spaces.

In the case of σ-finite measure spaces, the theorem classifies them up to an isomorphism, where an isomorphism is an equivalence class of measurable bijections $f$ with measurable inverse such that $f$ and $f^{-1}$ preserve measure 0 sets.

As explained in the article categories of measure theory, for a truly general, unrestricted statement for non-σ-finite measure spaces there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and σ-finite measure spaces should be replaced by Marczewski-compact strictly localizible measure spaces. For the sake of brevity, we refer to such objects simply as measure spaces below. (As explained in detail in the article categories of measure theory, the actual choice of a measure is immaterial for the discussion below, and all what really matters is the σ-ideal of sets of measure 0.)

In this unrestricted form, by the Gelfand-type duality for commutative von Neumann algebras, Maharam’s theorem also classifies isomorphism classes of localizable Boolean algebras (equivalently: measurable locales), abelian von Neumann algebras, and hyperstonean spaces (or hyperstonean locales). (In particular, the original formulation in the 1942 paper by Maharam uses measure algebras, i.e., Boolean algebras of measurable subsets modulo measure 0 subsets.)

Statement

Every measure space canonically decomposes as a coproduct (disjoint union) of ergodic measure spaces. Here a measure space $X$ is ergodic if the only subobjects of $X$ (i.e., equivalence classes of measurable subsets modulo measure 0 subsets) invariant under all automorphisms of $X$ are $\emptyset$ and $X$ itself.

Furthermore, an ergodic measure space $X$ is (noncanically, using the axiom of choice) isomorphic to $\mathfrak{c}\times 2^\kappa$, where $\kappa$ is 0 or infinite, and $\mathfrak{c}$ is infinite if $\kappa$ is infinite. Here the cardinal $\mathfrak{c}$ is known as the cellularity of $X$ and $\kappa$ is its Maharam type.

In particular, if $\kappa=0$, we get a classification of isomorphism classes of atomic measure spaces: they are classified by the cardinality $\mathfrak{c}$ of their set of atoms.

Otherwise, $\kappa$ is infinite, and we get a classification of isomorphism classes of ergodic atomless (alias diffuse) measure spaces: such spaces are isomorphic to $\mathfrak{c}\times 2^\kappa$, where $\mathfrak{c}$ and $\kappa$ are infinite cardinals.

For $\kappa=\aleph_0$, we get $2^\kappa\cong\mathbf{R}$, the real line. For $\kappa\gt\aleph_0$, we get non-σ-finite measure spaces, which occur naturally in stochastic processes.

Thus, a completely general object $X$ has the form

$$A\sqcup\coprod_\kappa \mathfrak{c}_\kappa\times 2^\kappa,$$

where $A$ is a discrete measure space, $\kappa$ runs over all infinite cardinals, $\mathfrak{c}_\kappa$ is an infinite cardinal or 0, and $\mathfrak{c}_\kappa\ne0$ only for a set of cardinals $\kappa$.

Such an object $X$ is a σ-finite measure space if and only if $A$ and $\mathfrak{c}_{\aleph_0}$ are countable and for every $\kappa\gt\aleph_0$ we have $\mathfrak{c}_\kappa=0$.

Relative version

There is also a relative version of Maharam’s theorem, which classifies morphisms in any of the equivalent categories considered above.

Observe that morphisms $2^\kappa\to2^\lambda$ exist if and only if $\kappa\ge\lambda$. For example, there are no morphisms from the terminal space $2^0$ (i.e., a singleton) to the real line $\mathbf{R}\cong 2^{\aleph_0}$, since the image of such a point is a measure 0 subset, whose preimage therefore cannot have measure 0. In the language of commutative von Neumann algebras, this translates to saying that there are no normal *-homomorphisms $L^\infty(\mathbf{R})\to\mathbf{C}$.

Observe also that we have a natural notion of locality for a measure space: a covering family is given by a family of measurable subsets whose essential supremum equals the entire space. (This is more than just an analogy to open covers in topological spaces: when translated to the language of locales, the two notions become identical.) In fact, thanks to the very special nature of measure spaces, it suffices to consider only disjoint covers, since every covering family as defined above can be refined by a disjoint family. Maharam’s theorem can be now formulated by saying that a

With these two observations in mind, we can succinctly formulate the relative Maharam theorem as follows: every morphism $f\colon X\to Y$ locally in $X$ and $Y$ is isomorphic to a morphism of the form $2^\kappa\to2^\lambda$, where $\kappa\ge\lambda$ and the map is given by projecting to the first $\lambda$ coordinates.

References

The original reference is

• Dorothy Maharam?, On homogeneous measure a lgebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942) 108-111. doi.

A modern exposition can be found in Chapter 33 (Volume 3, Part I) of

• David H. Fremlin, Measure Theory.