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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 19th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea Maharam's theorem states a complete classification of isomorphism classes of the appropriate [[category of measurable spaces]]. In the [[σ-finite]] case, the theorem classifies [[measure spaces]] up to an isomorphism. Here 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 spaces]] there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and [[σ-finiteness]] should be relaxed to a combination of Marczewski-compactness and strict localizibility. 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]], [[abelian von Neumann algebras]], and [[hyperstonean spaces]] (or [[hyperstonean locales]]). ## Statement Every object in one of the above equivalent categories canonically decomposes as a [[coproduct]] (disjoint union) of ergodic objects. Here an object $X$ is ergodic if the only subobjects of $X$ invariant under all automorphisms of $X$ are $\emptyset$ and $X$ itself. Furthermore, an ergodic object $X$ 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 (or diffuse) measure spaces: such spaces are isomorphic to $\mathfrak{c}\times 2^\kappa$, where $\mathfrak{c}$ and $\kappa$ are infinite cardinals. Thus, a completely general object $X$ has the form $$\coprod_\kappa \mathfrak{c}_\kappa\times 2^\kappa,$$ where $\kappa$ runs over 0 and all infinite cardinals, $\mathfrak{c}_\kappa$ is a cardinal that is infinite or 0 if $\kappa\ne0$, and $\mathfrak{c}_\kappa\ne0$ only for a set of $\kappa$. ## References The original reference is * [[Dorothy Maharam]], _On homogeneous measure a lgebras_, Proc. Nat. Acad. Sci. U.S.A. 28 (1942) 108-111. [doi](https://doi.org/10.1073/pnas.28.3.108). A modern exposition can be found in Chapter 33 (Volume 3, Part I) of * [[David H. Fremlin]], _[[Measure Theory]]_. <a href="https://ncatlab.org/nlab/revision/Maharam%27s+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Maharam%27s+theorem">current</a>

   Created:

   Idea

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

   In the σ-finite case, the theorem classifies measure spaces up to an isomorphism. Here 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 spaces there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and σ-finiteness should be relaxed to a combination of Marczewski-compactness and strict localizibility.

   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, abelian von Neumann algebras, and hyperstonean spaces (or hyperstonean locales).

   Statement

   Every object in one of the above equivalent categories canonically decomposes as a coproduct (disjoint union) of ergodic objects. Here an object $X$ is ergodic if the only subobjects of $X$ invariant under all automorphisms of $X$ are $\emptyset$ and $X$ itself.

   Furthermore, an ergodic object $X$ 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 (or diffuse) measure spaces: such spaces are isomorphic to $\mathfrak{c}\times 2^\kappa$, where $\mathfrak{c}$ and $\kappa$ are infinite cardinals.

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

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

   where $\kappa$ runs over 0 and all infinite cardinals, $\mathfrak{c}_\kappa$ is a cardinal that is infinite or 0 if $\kappa\ne0$, and $\mathfrak{c}_\kappa\ne0$ only for a set of $\kappa$.

   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.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 19th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a relative version: ## 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.) 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. <a href="https://ncatlab.org/nlab/revision/diff/Maharam%27s+theorem/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Maharam%27s+theorem/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Maharam%27s+theorem">current</a>

   Added a relative version:

   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.)

   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.

   diff, v2, current