Spectrum of $L^\infty(X,\mu)$

Here is a description of the spectrum of $L^\infty([0,1];\mu)$ for an arbitrary Borel measure $\mu$ on $[0,1]$.

Consider the following poset, which I call $P$ :

• The objects of $P$ are decompositions $\mathbf X=\{X_1,\ldots, X_n\}$ of $[0,1]$ into finitely many $\mu$-measurable sets $[0,1]=X_1\cup X_2\cup\ldots\cup X_n$,  $X_i\cap X_j=\emptyset$. Two decompositions $\mathbf X$ and $\mathbf Y$ are declared equal is there exists a permutation $\sigma$ such that $X_i=Y_{\sigma(i)}$ up to a $\mu$-measure zero set.
• The partial order on $P$ is given by refinement: $\mathbf X \prec \mathbf Y$ if
$Y_1=X_1\cup\ldots \cup X_{n_1}$, $Y_2=X_{n_1+1}\cup\ldots \cup X_{n_2}$, $\ldots$ (up to permutation and $\mu$-measure zero sets)

Note that the poset $P$ is filtered: given a finite set $\mathbf X_1, \mathbf X_2, \ldots, \mathbf X_n$ of elements of $P$, there is always a common refinement, i.e., an element $\mathbf X\in P$ such that $\mathbf X\prec \mathbf X_i\\,\forall i$

Given a $\mu$-measurable subset $X\subset [0,1]$, let me denote by $|X|_ \mu \subset [0,1]$ the $\mu$-adherence of $X$: $$ |X|_ \mu:=\{x\in [0,1]: \forall \varepsilon>0\quad \mu(X\cap B_{x,\varepsilon})>0\}, $$ where $B_{x,\varepsilon}$ denotes the ball of radius $\varepsilon$ around the point $x$. Given $\mathbf X=\{X_1,\ldots,X_n\} \in P$, we also write $|\mathbf X|_ \mu$ for the disjoint union $$|\mathbf X|_ \mu:=|X_1|_ \mu\sqcup\ldots\sqcup|X_n|_ \mu.$$

Note that if $\mathbf X \prec \mathbf Y$, then there is a natural projection map $|\mathbf X|_ \mu \twoheadrightarrow |\mathbf Y|_ \mu$.

Given the above preliminaries, the spectrum of $L^\infty([0,1];\mu)$ is given by the inverse limit of the functor $P\to Top, \mathbf X\mapsto |\mathbf X|_ \mu$:

$$Spec\big(L^\infty([0,1];\mu)\big) =\quad \underset{\mathbf X\in P}{\underset\leftarrow\lim} |\mathbf X|_ \mu$$

I'm recording here some references on the object $\tilde X = \mathrm{Spec}(L^\infty(X,\Sigma,\mu))$ (many of which were communicated to me recently by Balint Farkas), assuming for sake of simplicity that $\mu$ is a probability measure to avoid some technicalities. As the references below show, this object has been discovered and used multiple times in the literature, but lacks a standardised name.

The earliest reference I know of that uses this space is

Halmos, Paul R., On a theorem of Dieudonne, Proc. Natl. Acad. Sci. USA 35, 38-42 (1949). ZBL0031.40701.

who calls it the "Kakutani space" of $X$. Another early reference (referring to $\tilde X$ as a "perfect measure space") is

Segal, I. E., Equivalences of measure spaces, Am. J. Math. 73, 275-313 (1951). ZBL0042.35502.

and the space is referred to as the "Gelfand space" of $X$ and used to relate ergodic theory with topological dynamics in

Ellis, Robert, Topological dynamics and ergodic theory, Ergodic Theory Dyn. Syst. 7, 25-47 (1987). ZBL0592.28015.

An equivalent construction of the space (based on the Loomis-Sikorski theorem) also appears in

Doob, Joseph L., A ratio operator limit theorem, Z. Wahrscheinlichkeitstheor. Verw. Geb. 1, 288-294 (1963). ZBL0122.36302.

and implicitly in

Fremlin, D. H., Measure theory. Vol. 3. Measure algebras, Colchester: Torres Fremlin (ISBN 0-9538129-3-6/pbk). 693 p., 13 p. (2004). ZBL1165.28002.

The space is also discussed in Chapter 12 of the recent text

Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer, Operator theoretic aspects of ergodic theory, Graduate Texts in Mathematics 272. Cham: Springer (ISBN 978-3-319-16897-5/hbk; 978-3-319-16898-2/ebook). xviii, 628 p. (2015). ZBL1353.37002.

(who call it the "Stone model" of $X$) and in a recent preprint of Jamneshan and myself we refer to it as the "canonical model" and rely on it to perform various product-type constructions on abstract measure-preserving systems. As we note in that paper, it behaves in many ways like the Stone-Cech compactification of locally compact Hausdorff spaces.

Some key properties of this space (discussed for instance in my paper with Jamneshan):

• $\tilde X$ is a Stonean space (compact Hausdorff and extremally disconnected). In particular, by a result of Gleason, it is projective in the category of compact Hausdorff spaces: given any surjective continuous map $\pi: Y \to Z$ on compact Hausdorff spaces, any continuous map from $\tilde X$ to $Z$ can be continuously lifted to $Y$. One can also identify $\tilde X$ with the Stone space of the probability algebra of $X$ (the measurable sets modulo the null sets). In particular, by the Loomis-Sikorski theorem, the probability algebra of $X$ is isomorphic to the Baire sets of $\tilde X$ modulo the meager sets.

• The map $X \mapsto \tilde X$ can be viewed as a covariant functor from the category of probability spaces (or an abstraction of this category which we call the category of opposite probability algebras) to the category of compact Hausdorff probability spaces. In fact it is left-adjoint to the forgetful functor from compact Hausdorff probability spaces to opposite probability algebras (much as the Stone-Cech compactification is left-adjoint to the forgetful functor from compact Hausdorff spaces to locally compact Hausdorff spaces). Related to this, $\tilde X$ is "universal" amongst all compact Hausdorff spaces whose probability algebra is isomorphic to that of $X$, in the same way that the Stone-Cech compactification is universal amongst all compactifications of a locally compact Hausdorff space.

• There is a natural lift $\tilde \mu$ of $\mu$ to $\tilde X$, using the Baire $\sigma$-algebra on $\tilde X$. A Baire set in $\tilde X$ is $\tilde \mu$-null if and only if it is meager. If $X$ is itself a compact Hausdorff space (with the Baire $\sigma$-algebra), then $\mu$ is Radon and there is a continuous probability-preserving map from $\tilde X$ to $X$, whose image is the support of $\mu$.

• One has an isomorphism $C(\tilde X) = L^\infty(\tilde X)$. Thus $\tilde X$ enjoys a strong version of Lusin's theorem that we refer to as the "strong Lusin property": every bounded Baire-measurable function on $\tilde X$ is equal outside of a null set (or equivalently, a meager set) to a unique continuous function. Similarly, any Baire-measurable map from $X$ to a compact Hausdorff space $K$ can be uniquely identified with a continuous map from $\tilde X$ to $K$.