Which sigma-ideals in a sigma-algebra are ideals of null sets?

Question

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space but are just told which sets are null - what can we say about the measure space? Note that, for example, membership in $L^\infty$ depends only on the questions of which sets are null.

Concretely, let $ \left (X, \mathcal{B} \right ) $ be a measurable space. A $\sigma$-ideal $\mathcal{I}$ is a subset of $\mathcal {B}$ which is closed under countable unions, contains the null set, and such that whenever $I \in \mathcal{I}$ and $B \in \mathcal{B}$, $I \cap B \in \mathcal{I}$. The standard example of $\sigma$-ideal is the family of null sets in a measure space $\left (X, \mathcal{B}, \mu \right) $.

Question: Given an arbitrary $\sigma$-ideal in a $\sigma$-algebra, when is it the family of null sets for some measure on the space?

I have not been able to find an answer to this question that does not amount to a tautology, for example, assuming that there exists a positive-definite function on $\mathcal{B}$ which essentially satisfies the axioms of a measure.

My approach to the question so far has been to attempt to define an analogue of $L^\infty$ for this case: for a measurable $f:X \to \mathbb{C}$, define the essential supremum $ \left \|f \right \|_\infty$ to be the infimum over all positive numbers $M$ such that $|f(x)| \le M$ outside of some set in $\mathcal{I}$, and let $L^\infty (X, \mathcal{B}, \mathcal{I})$ be the space of all functions with finite essential supremum.

It is easy to check that $L^\infty (X, \mathcal{B}, \mathcal{I})$ is always a Banach space, and by defining the $*$ operation to be conjugation as usual, it in fact becomes a $C^*$-algebra. If we knew that it is a Von-Neumann algebra (actually, a $W^*$-algebra, meaning it has a predual), it would be easy to conclude that $\mathcal{I}$ is the family of null sets for a measure, since then our space would be isomorphic to $L^\infty (Y, \mathcal{S}, \mu)$ and this isomorphism would necessarily take characteristic functions to characteristic functions.

From here, I can think of two ways to proceed:

1. Define also an analogue of $L^1$ in this case: since Radon-Nikodym allows us to identify $L^1$ with the space of all finite measures which are absolutely continuous wrt to a given measure, we define $L^1(X,\mathcal{B},\mathcal{I})$ to be the set of all measures which are zero on all sets in $\mathcal{I}$. The first problem with this is that in general, $(L^1)* \neq L^\infty$. This can be remedied by making a very reasonable assumption on $\mathcal{I}$, which I can say more about if needed, but even then, I don't see how to proceed.

2. Find a representation of $L^\infty(X,\mathcal{B},\mathcal{I})$ on a separable Hilbert space. However, I don't know how to construct a suitable Hilbert space without some sort of tautological assumption on the $\sigma$-algebra of the style I have written above.

Any reference or ideas would be greatly appreciated.

Answer 1

First of all, one should mention that not every triple (X,B,μ) (i.e., what is often called a measure space) satisfies the property that its C*-algebra of bounded functions is a von Neumann algebra (= W*-algebra) or that the map L^∞→(L_1)* is an isomorphism. One has to impose additional conditions to ensure that this is true.

If we do not assume such a condition, Joseph Van Name's answer solves the problem.

However, if one is interested in a condition that would guarantee that (X,B,I) (the σ-ideal I is often known as a measure class) comes from a triple (X,B,μ) whose algebra of bounded functions is a von Neumann algebra (and pretty much all of measure theory happens in this restricted setting because otherwise almost every nontrivial theorem would fail), then an answer was given in 1951 by Irving Segal in his paper “Equivalences of measure spaces”.

Segal proved that several seemingly unrelated properties are equivalent for a triple (X,B,I) (i.e., a measure space equipped with a measure class). Two of these properties have already been named: L^∞(X,B,I) is a von Neumann algebra and the canonical map L^∞(X,B,I)→L^1(X,B,I)* is an isomorphism (i.e., the Riesz representation theorem is satisfied). Among other equivalent properties one can find the Radon-Nikodym theorem, the Hahn decomposition theorem, and the fact that the Boolean algebra B/I is complete. The full list and additional details can be found in the answer https://mathoverflow.net/a/20820/402.

Such triples (X,B,I) are known as localizable measurable spaces and they can be organized into a category whose morphisms (X,B,I)→(Y,C,J) are equivalence classes of measurable maps X→Y such that the preimages of elements of J belong to I (this true in the complete case, in the general case a slightly more complicated definition is necessary, as described in the link).

This category turns out to be contravariantly equivalent to the category of commutative von Neumann algebras (i.e., a version of Gelfand duality is true for von Neumann algebras), which further stresses the importance of the localizability condition. More information about this category and further links can be found in this answer: https://mathoverflow.net/a/49542/402.

Answer 2

As G. Rodrigues stated, the problem of finding measures on a $\sigma$-algebra $(X,\mathcal{M})$ with a $\sigma$-ideal of null sets $I$ reduces to the problem of finding measures on the Boolean algebra $\mathcal{M}/I$. It turns out that the problem of finding measures $\mu$ on $\sigma$-complete Boolean algebras $B$ with $\mu(a)=0$ iff $a=0$ only appears to be more general than the problem of finding measure on $\sigma$-algebras $(X,\mathcal{M})$ with $\sigma$-ideals $I$. This is because every $\sigma$-complete Boolean algebra is isomorphic to a quotient of a $\sigma$-algebra by a $\sigma$-complete ideal.

$\mathbf{Theorem}$(Loomis-Sikorski) Suppose that $B$ is a $\sigma$-complete Boolean algebra. Then there is a $\sigma$-algebra $(X,\mathcal{M})$ and a $\sigma$-complete ideal $I\subseteq\mathcal{M}$ such that $\mathcal{M}/I\simeq B$.

The paper 1 by Thomas Jech gives several characterizations of the $\sigma$-complete Boolean algebras $B$ such that there is some measure $\mu$ that makes $(B,\mu)$ a $\sigma$-finite measure algebra. I will state a couple of those characterizations here and I would encourage the reader to look at Jech's paper for the other characterizations and more information.

If $B$ is a Boolean algebra, then a cellular family is a subset $r\subseteq B\setminus\{0\}$ such that if $a,b\in r,a\neq b$, then $a\wedge b=0$. A partition of a Boolean algebra $B$ is a cellular family $p\subseteq B\setminus\{0\}$ with $\bigvee p=1$. A Boolean algebra $B$ satisfies the countable chain condition if every partition of $B$ is countable. A Boolean algebra $B$ is weakly distributive if whenever $p_{n}$ is a partition of $B$ for all natural numbers $n$ there is a partition $p$ of $B$ where if $a\in p,n\in\mathbb{N}$, then the set $\{b\in p_{n}|a\wedge b\neq 0\}$ is finite.

A measure algebra is a pair $(B,\mu)$ such that $B$ is a $\sigma$-complete Boolean algebra and $\mu:B\rightarrow[0,\infty]$ is a mapping such that $\mu(a)=0$ iff $a=0$ and $\mu(\bigvee_{n}a_{n})=\sum_{n}\mu(a_{n})$ whenever $a_{n}\wedge a_{m}=0$ for $m\neq n$.

$\mathbf{Theorem}$ Let $B$ be a $\sigma$-complete Boolean algebra. Then the following are equivalent.

1. There is a measure $\mu$ such that $(B,\mu)$ is a measure algebra with $\mu(1)=1$.

2. $B$ is weakly distributive and $B^{+}$ is the union of a sequence $(C_{n})_{n\in\mathbb{N}}$ that satisfies the following properties.

   i. For all $n$, there is some integer $K(n)$ such that every cellular family in $C_{n}$ has at most $K(n)$ elements.

   ii. If $a_{n}\not\in C_{n}$ for all $n$, then $\overline{\textrm{Lim}}_{n}a_{n}=\bigwedge_{n}\bigvee_{k\geq > n}a_{k}=0$.

3. $B$ is weakly distributive and $B\setminus\{0\}$ is the union of a sequence $(C_{n})_{n}$ of subsets of $B$ where for all $n$

   i. there is an integer $K(n)$ where every cellular family in $C_{n}$ has at most $K(n)$ elements

   ii. If $a\vee b\in C_{n}$, then $a\in C_{n+1}$ or $b\in C_{n+1}$.

1 Jech, Thomas (CZ-AOS) Algebraic characterizations of measure algebras (English summary) Proc. Amer. Math. Soc. 136 (2008), no. 4, 1285–1294.

Answer 3

Your question has already been excellently answered from two points of view:

(a) looking at the quotient $\sigma$-algebra (measurable sets modulo null sets): when is it a measure algebra? [Joseph Van Name, citing Jech]

(b) looking, as you suggested, at the $C^*$-algebra of bounded measurable functions modulo the almost everywhere null functions: when is it a von Neumann algebra? [Dmitri Pavlov, citing I. Segal]

Case (a) is more specialized than case (b), which in turn is not the most general nontrivial case.

After two sections with definitions and basic facts (about measure spaces and $*$-operator algebras), the final section (the core of this answer) considers the missing cases (so someone might want to skip to the final section).

1: Four levels of measure spaces

In terms of measure spaces $(X,\mu)$ (where the $\sigma$-algebra is the domain of definition of $\mu$ and the $\sigma$-ideal corresponds to $\mu$-null sets) the four cases are, in increasing level of generality:

$\bullet$ (a) finite or equivalently $\sigma$-finite $\mu$ (using a countable partition of $X$ in sets $X_i$ with $\mu$-finite measure, rescaling $\mu$ by a suitable factor $a_i$ on each $X_i$ gives a convergent sum of the $a_i\mi(X_i)$ hence a finite measure).

$\bullet$ (b') strictly localizable case. The measure space is direct sum of finite measure spaces, i.e. there is a (possibly uncountable) partition of $X$ in sets $X_i$ of finite $\mu$-measure such that: $\mu(Y)$ is defined iff each $\mu(Y\cap X_i)$ is defined, and then $\mu(Y)$ is the sum of the $\mu(Y\cap X_i)$. [Typical examples are Radon measures, the only measures considered by Bourbaki.]

$\bullet$ (b) localizable case. The boolean algebra of measurable sets modulo null sets is complete (not only $\sigma$-complete); equivalently: the vector lattice of real measurable functions modulo null functions is conditionally Dedekind complete (not only $\sigma$-complete); equivalently: the $C^*$-algebra of complex measurable functions modulo null functions is a $W^*$-algebra (it has a predual as Banach space, equivalently: it has a faithful representation in a Hilbert space as von Neumann algebra [i.e. as a self-adjoint set of bounded linear operators which is the commutant of its commutant]).

The equivalent conditions above only depend upon the "abstract" quotient algebra (measurable sets modulo null sets); on the contrary, the concept of strict localizability depends upon the "concrete" realization as algebra of sets. This is the only difference between the two concepts: localizability is about "measure algebras" (no $X$ is present), strict localizability is about "measure spaces" (a structured $X$).

Assuming that $\mu$ is complete (all subsets of a $\mu$-null set are in the domain of $\mu$) and "semi-finite" (there are no sets of infinite measure with only null sets as subsets of finite measure), the following conditions are equivalent to localizability: Radon - Nykodym (finite measures absolutely continuous with respect to $\mu$ are represented by elements of $L^1$); $L^\infty$ is the Banach dual of $L^1$ by means of the natural duality $(f,g)\mapsto\int_X fgd\mu$.

Under the same two assumptions, the existence of a lifting (a subalgebra of the bounded measurable functions that has exactly one representant for each element of $L^\infty$, or equivalently the analogue for $L^\infty$ as vector lattice, or equivalently the analogue for the Boolean algebra of measurable sets) is equivalent to strong localizability and so it is a strictly stronger condition. See Fremlin's book (free at Fremlin's home page) for more equivalent conditions for (strict) localizability, and examples to show distinctions of the two concepts (and strangeness of the incomplete or non semi-finite cases).

$\bullet$ (c) as "general" case, one could take an arbitrary $\mu$. Then each $\sigma$-ideal in a $\sigma$-algebra gives a $\mu$. null on the $\sigma$-ideal and infinite outside (as noted by Stefan Waldmann). Avoiding such uninteresting cases, the most general interesting case the semi-finite measures; each $\mu$ produces an associated semi-finite measure $\mu'$ (the largest semi-finite $\mu'\leq\mu$): $\mu'(Y)$ is the sup of the finite $\mu(Y')$ with $Y'\subseteq Y$.

Example: the semi-finite measure associated to a $\mu$ that takes only $0,\infty$ as values is the identically zero measure; hence, in general, the associated semi-finite measure gives a different boolean algebra of measurable sets modulo null sets and a different $L^\infty$.

Strictly localizable (in particular, $\sigma$-finite) measures are semi-finite. Completing a measure (by declaring measurable each symmetric difference of a set in the domain of $\mu$ and a subset of a $\mu$-null set, and then extending $\mu$ in the only possible way) does not change (up to canonical isomorphisms) the (metric) boolean algebra of measurable sets modulo null sets, nor the space of step functions modulo null functions and its completions (the spaces $L^p$), nor the semi-finiteness (localizability, $\sigma$-finiteness, finiteness) status of the measure.

As noted by Dmitri Pavlov, almost all notrivial theorems require localizability, and in fact are often equivalent to it. So localizability is in a sense the most central among the possible conditions on a measure space. There are a few exceptions. First: theorems requiring conditions stronger than localizability. Some cases are easily equivalent to finiteness (the constants are integrable, uniform convergence for integrable functions implies convergence of the integrals and more generally $L^q$ is contained in $L^p$ when $0\leq p<q\leq\infty$), or to $\sigma$-finiteness (there is a everywhere strictly positive integrable function). As noted above, existence of lifting is equivalent to strict localizability. Second: there are interesting theorems that do not require localizability. Two examples: for conjugate exponents $1<pq=p+q<\infty$, the natural duality by integration gives $L^q$ as Banach dual of $L^p$. Also the Grothendieck property of $L^\infty$ holds true even for non semi-finite $\mu$.

2: Four levels of operator algebras / abstract boolean algebras

Commutative unital $C^*$-algebras (or equivalently Banach lattices with a fixed "strong order unit" $u$ [each $f$ satisfies $|f|\leq nu$ for some scalar $n$] and as norm the Minkowski functional of the lattice interval $[-u,u]$) are in Gelfand duality with compact T$_2$ spaces (or equivalently their lattice of open sets: a complete Heyting algebra with the obvious lattice translations of compactness, T$_1$ [co-atomistic] and regularity or equivalently normality). The points of the space are the maximal i.e. primitive i.e. closed prime ideals of the $C^*$-algebra, or equivalently the extreme points of the compact convex set of states on the $C^*$-algebra i.e. the two-valued Radon probability measures on the compact T$_2$ space; the topology is the one induced by the compact convex set of all states (with the weak topology of the duality with the $C^*$-algebra of continuous functions on the compact T$_2$ space).

The 0-dimensional case is that of the Stone duality between Boolean algebras (of idempotents i.e. projections i.e. direct decompositions of the $C^*$-algebra or vector lattice) and Boolean spaces (the lattice of open sets is the lattice of ideals of the boolean algebra of clopen sets, i.e. the Heiting algebra is an algebraic lattice, besides having the preceding properties). Points of the space are, as extreme states, the two-valued [a.k.a. "dispersion free"] finitely additive probability measures on the Boolean algebra, or equivalently the maximal i.e. prime ideals of the Boolean ring i.e. lattice.

$\bullet$ What matters here is an even more specialized subcase: that of countably complete boolean algebras. They are the abstract $\sigma$-algebras, which need not be isomorphic to $\sigma$-algebras of sets (see below). The corresponding compact T$_2$ spaces are called "basically disconnected" or "$\sigma$-stonian": the closure of a open $F_\sigma$ set is again open (in presence of such an axiom, 0-dimensionality is equivalent to complete regulatity, or regularity, or semi-regularity [the sets that are the interior of their closure give a basis for the topology]; the open $F_\sigma$ sets are then the co-zero sets of continuous real functions, so the axiom plus the preceding equivalent conditions is equivalent to: the closure of co-zero sets are exactly the clopen sets, which form a basis for the topology). The corresponding $C^*$-algebras (of continuous complex functions) are the Rickart$^*$-algebras: unital $C^*$-algebras such that the (right) annihilator of an element is generated (as right ideal) by a idempotent (and then it is generated by a unique self-adjoint idempotent [projection], and the projections with the divisibility order are naturally isomorphic to the Boolean algebra of clopen sets; note that such a alegebra is countably complete, but infinite countable Sup and Inf operations are generally not the set joins and meets as subsets of the T$_2$ space, i.e. the algebra is not a sub $\sigma$-algebra of subsets of $X$). Another equivalent way to look at such objects: vector lattices with a strong order unit which are Dedekind $\sigma$-complete (each bounded increasing sequence has a Sup); a commutative unital $C^*$-algebra is Rickart iff the ordered vector space of self-adjoint elements is such a vector lattice (with the 1 of the algebra as strong order unit), and conversely each such vector lattice comes from a canonically unique Rickart$^*$-algebra (the complexification of the vector lattice with its unique ring structure that makes 1 the given strong order unit and has the ring squares as positive cone).

Such vector lattices are studied in books about vector lattices: Kantorovich - Vulikh - Pinsker; Meyer-Nieberg [Banach lattices]; Schaefer [Banach lattices and positive operators]; Luxemburg and Zaanen [Riesz spaces], ...

Their corresponding complex algebras were studied, also in a non-commutative setting, by Rickart and Kaplansky [rings of operators], then Berberian [Baer$^*$-rings] and many others (but with no further specialized books).

Once Rickart$^*$-algebras are introduced, the subclasses that are important for the relation with measure algebras and spaces are the following (to be defined below):

           /
Rickart   .----  Baire / Sigma
  /|\      \        /|\
   |                 |
   |        /        |
Baer (AW)  .----  von Neumann (W)
            \

Again four cases, but they are not in perfect correspondence with the four "measurable" cases (the "measurable" cases are chain-ordered by implication, the present cases form a four element boolean algebra). But a good correspondence exists, see the last part. But before the correspondence, terms must be defined.

$\bullet$ AW$^*$-algebras are the $C^*$-algebras that form a Baer ring: the (right) annihilator of an arbitrary subset (not only of single elements) is generated by an idempotent. Equivalently (even in a non-commutative setting), a Rickart$^*$-algebra where the lattice of annihilators is complete (and not only countably complete). In terms of Gelfand duality, in the commutative case they correspond to complete (and not only countably complete) boolean algebras, and Stonian compact T$_2$ spaces: the closure of an open set is again open ("extremally disconnected space"; in presence of such an axiom, 0-dimensionality, complete regularity, regularity and semi-regularity are equivalent; so the axiom plus the equivalent conditions exactly says: the closures of open sets are exactly the clopen sets, which give a basis for the topology). A theorem of Gleason says that the Stonian spaces are precisely the projective objects in the category of compact T$_2$ spaces (the theorem really gives the projective objects in the category of T$_2$ spaces, and gives a projective cover for each object); by Gelfand duality, one obtains the injective objects among unital commutative $C^*$-algebras (AW$^*$ algebras) and injective objects among boolean algebras (complete boolean algebras). Dedekind completion by cuts of a Boolean algebra (resp. vector lattice) gives the injective envelope of a boolean algebra (resp. vector lattice).

$\bullet$ $C^*$-algebras with a Banach predual were recalled above as abstract characterization (due to Sakai) of von Neumann algebras. From the present point of view, another charaterization (due to Kadison) is important: AW$^*$-algebras with sufficiently many normal states. The definitions of "state", "normal" and "sufficiently many" follow.

A state on a $C^*$-algebra is a linear functional $\phi$ such that $\forall x,\phi(xx^*)\geq0$ and $\phi(1)=1$; they are automatically continuous of norm 1 (in fact they are the positive unit preserving linear functionals on the ordered vector space with fixed unit of self-adjoint elements).

(In the definition of states, the restriction of the domain of a state to self adjoint elements is used to avoid complications in the noncommutative case but it is not important in the case of interest here: commutative case i.e. the self adjoint elements for a vector lattice).

By Gelfand duality the states on commutative $C^*$-algebras precisely correspond to Radon measures on the compact T$_2$ spectrum of the algebra; so in the 0-dimensional case they precisely correspond to finitely additive probability measures on the boolean algebra of clopen sets. Normality of a state on a AW$^*$-algebra means that it is continuous for monotone convergence of nets (not only sequences) in the vector lattice of self-adjoint elements, i.e. it is completely additive (not only countably additive) on the complete boolean algebra of clopen sets, i.e. that the corresponding Radon measure is null on the $\sigma$-ideal of first category sets (i.e. every closed nowhere dense set has measure zero).

One says that there are sufficiently many such normal states when $\forall x\neq0$ in the AW$^*$-algebra there is a normal state $\phi$ such that $\phi(xx^*)\neq0$ i.e. for each nonzero projection $x$ there is such a $\phi$ i.e. for each nonempty clopen $A$ (hence each nonempty open $A$) there is a normal Radon measure which is nonzero on $A$.

The connection between Sakai's characterization and Kadison's one: the Banach predual of a von Neumann algebra is unique; it is the vector space generated by the normal states with the unique norm that makes them of norm 1.

The finite dimensional $C^*$-algebras are exactly the finite direct sums of matrix $*$-algebras over the complex $*$-field. The commutative case corresponds to compact discrete i.e. finite spaces. They are exactly the W$^*$ which are separable as $C^*$-algebras [Banach spaces]. The world "separable" for W$^*$-algebras is used in a weaker sense: there is a faithful representation as von Neumann algebra on a separable Hilbert space iff the predual is separable as Banach space iff the algebra is separable [not in the norm topology, but] in the weak operator topology iff the algebra is "countably generated" [there is a countable subset not contained in any proper W$^*$-subalgebra] and "orthoseparable" a.k.a "c.c.c." a.k.a. "countably decomposable" [every collection of orthogonal nonzero projections is at most countable].

Note that in presence of orthoseparability countable completeness and completeness coincide, and so also Rickart and AW$^*$ coincide; in particular this happens for algebras with a faithful $C^*$ representation in a separable Hilbert space. Standard example for the distinction between countably complete and complete (i.e. Rickart vs. AW$^*$): the $\sigma$-algebra of countable / co-countable subsets of a uncountable set (and its Hilbertian analogue for a irreducibly non-commutative example).

$\bullet$ Finally, the $\Sigma^*$ $C^*$-algebras introduced by Davies and further studied by Pedersen (also in his book "$C^*$-algebras and their automorphism groups"), Dang Ngoc Nghiem, Kehlet. [The Baire$^*$ algebras studied by Pedersen and Kehlet might be more general than $\Sigma^*$ algebras, but in the commutative case both concepts are equivalent to: commutative Rickart$^*$-algebras with sufficiently many $\sigma$-additive measure on the Boolean $\sigma$-algebra of projections.]

Kadison's characterization of von Neumann algebras uses complete additivity; the present class of algebras is analogously characterized by $\sigma$-additivity. So $C^*$-algebras that are weakly sequentially closed in at least one W$^*$-algebra are precisely the sequentially monotone complete $C^*$-algebras with sufficiently many countably additive states.

In the commutative case they are exactly the Rickart$^*$-algebras corresponding to set-representable Boolean $\sigma$-algebras, i.e.: countably complete Boolean algebras with sufficiently many two-valued countably additive probability measures; each sufficient set of such measures gives the points for a representation as $\sigma$-algebra of sets, and conversely; the largest such representation uses all the two-valued countably additive probability measures as points, and it is strictly smaller [except the finite case] than the Stone representation, which uses all finitely additive two-valued probability measures and gives a representation that does not transform countable boolean operations into set operations.

In fact, starting from an arbitrary countably complete Boolean algebra $B$ and taking its associated compact basically diconnected space, $B$ is represented as homomorphic image of a $\sigma$-algebra of sets (Loomis - Sikorski): one takes the Baire $\sigma$-algebra of the space (the $\sigma$-algebra generated by the cozero sets i.e. the smallest $\sigma$-algebra that makes measurable all continuous real functions) and factors out the meager (first category) sets; the representation means that each Baire measurable set is the symmetric difference of a unique clopen set with a meager set (this is true for cozero sets by the "basically disconnected" hypothesis, and then one extends this to the generated $\sigma$-algebra). When sufficiently many two-valued $\sigma$-probabilities exists (and exactly only in that case), then one can restrict the representation set so that no more factoring out of meager sets is needed. The algebra of Borel sets in the real line modulo the $\sigma$-ideal of meager sets is the typical example of a orthoseparable ($\sigma$-)complete Boolean algebra with no nonzero countably additive semi-finte measure; the corresponding AW$^*$-algebra is not W$^*$ (Dixmier's example).

3: Correspondence between the measurable cases and the operator algebra cases.

What are the kinds of $C^*$-algebras obtained as $L^\infty$ in measurable spaces (with a fixed $\sigma$-ideal) of the various kinds?

$\bullet$ Since every countably complete boolean algebra is the quotient of a $\sigma$-algebra of sets by a $\sigma$-ideal, the most general (not so interesting) case is exactly given by Rickart$^*$-algebras. Moreover, among all spaces that give a predetermined Rickart$^*$-algebra i.e. equivalently a given quotient Boolean $\sigma$-algebra $B$, one can always choose the "universal" (largest faithful) one, the $\sigma$-stonian space $X$ of $B$, with the Baire $\sigma$ algebra and the $\sigma$-ideal of meager sets.

$\bullet$ At the opposite extreme, one has the $L^\infty$ of a finite (or equivalently $\sigma$-finite) measure. These $L^\infty$ are exactly the orthoseparable W$^*$-algebras. More precisely, one has [dual] equivalence between: commutative (Rickart or equivalently) AW$^*$-algebras with a fixed faithful normal state; Boolean $\sigma$-algebras with a fixed countably additive faithful probability measure; ($\sigma$-)Stonian compact T$_2$ spaces with a fixed Radon measure where each nonempty (cl)open set has nonzero measure and each meager set has zero measure.

If one wants to forget the normal state / measure, the complex algebras are exactly the orthoseparable W$^*$-algebras (not every orthoseparable Rickart i.e. AW$^*$ has such a faithful state: it has one iff it is W$^*$). Not every (orthoseparable and weakly distributive) Boolean $\sigma$-algebra has such a faithful $\sigma$-probability; Jech's paper gives conditions for this (necessary and sufficient conditions, when taken together with the classical orthoseparability and weakly distributivity known to be necessary since at least von Neumann, 1937). However, such conditions are more complicated and less unexpected than the W$^*$-condition for a orthoseparable $L^\infty$ (existence of a Banach predual; $*$-representability as a self-bicommuting $*$-subalgebra of continuous linear operators on a Hilbert space).

$\bullet$ Localizable and strictly localizable measure spaces give exactly the same $L^\infty$ algebras, i.e. exactly the commutative W$^*$-algebras (not necessarilly orthoseparable, i.e. a separating family of normal states might be necesarilly uncountable). As above one can equivalently represent these by special kinds of Stonian (compact T$_2$ extremally disconnected) spaces $X$ (they are called hyperstonian when sufficiently many normal Radon measures exist); what are the differences with the representation by measurable spaces instead?

The representation with (strictly) localizable measure spaces has the advantage that the condition on the measure is natural in such a context. Besides, the "separable" ones (in the von Neumann algebra sense, not in the $C^*$-algebra sense) permit representation with only a few models: the atomic models are the finite dimensional cases and $l^\infty$ (the counting measure on a countable set), the atomless case is $L^\infty[0,1]$ with the Lebesgue measure (or any other diffuse measure on a Polish space, they all give isomorphic $L^\infty$), and finally the direct sum of a atomic case with the atomless case. In general one has Maharam's representation theorem (see Fremlin's chapter 22 in the handbook of boolean algebras, volume 3). The disadvantage is that the measure space that represents a given $L^\infty$ is not unique up to isomorphisms; is is unique only up to a equivalence that kills the $\sigma$-ideal of measure zero sets. The topological representation has the opposite advantages and disadvantages: it is unique up to canonical isomorphisms (i.e. in the above equivalence class of measurable spaces with fixed $\sigma$-ideal one has only a Stonian representative (with the ideal of meager sets), unique up to canonical homeomorphism), but none of the "natural" measurable spaces has such a form (since the natural ones do not depend upon the existence of nonprincipal ultrafilters, but infinite Stonian spaces depends upon this) and the "normality" condition for measures on Stonian spaces is instead abnormal in ordinary contexts (Lebesgue measure does not satisfy this, and on the real line there is a sort of half-duality between "small in measure" and "small in topology": see Oxtoby, measure and category).

$\bullet$ Finally, there are the $L^\infty$ for semi-finite but not necesarilly localizable cases. These are the special kinds of Rickart$^*$-algebras considered by Davies et. al.; a $sigma$-complete (but not necessarilly complete) generalization of W$^*$-algebras. Again, one has dual advantages and disadvantages with the representations by measure spaces and $\sigma$-stonian spaces with sufficiently many normal measures. Again, it is quite intricate the characterization from the point of view of semi-finite measure algebras (they are the $\sigma$-complete Boolean algebras such that every nonzero element $x$ contains a nonzero $b$ such that the Boolean algebra $[0,b]$ is a finite-measure algebra, using Jech characterizations or any previous ones). Again, much more unexpected and plain is the characterization as: $L^\infty$ is weakly sequentially closed in a commutative von Neumann algebra.

$\bullet$ Most of this answer was a complement to Dmitri Pavlov's answer. So I end instead with a complement to Joseph Van Name's answer.

There is an alternative to characterizations of finite-measure algebras in the style of Jech's paper (and Fremin chapter, and so on).

For example, Jech uses a "filtration" of the Boolean algebra $B$ by considering the sets $B_n=\{b\in B\mid 2^n\mu(b)\geq 1\}$ and properties of this family that can be expressed without reference to $\mu$. Then one postulates the existence of such a family of sets with such properties, and from this one constructs a $\mu$.

One can use also other kinds of structures induced by $\mu$ such that the existence of such a structure with suitable axioms is equivalent to existence of $\mu$. What seems to me a very natural structure is the preorder relation "$a$ is more likely than $b$" defined by $\mu(a)\geq\mu(b)$ in presence of $\mu$. In fact such a relation is used, in a finitely additive context, to study "subjective probability" since its beginnings (a survey: http://projecteuclid.org/euclid.ss/1177013611 ).

C. H. Kraft, J. W. Pratt, A. Seidenberg for finite boolean algebras, C. Villegas for atomless Boolean $\sigma$-algebras, then R. Chuaqui, J. Malitz, M. G. Schwarze (see for example http://www.ams.org/journals/tran/1983-279-02/S0002-9947-1983-0709585-3/S0002-9947-1983-0709585-3.pdf and http://www.ams.org/journals/proc/1989-105-02/S0002-9939-1989-0935109-2/S0002-9939-1989-0935109-2.pdf ) and others extracted "qualitative" axioms (not involving real numbers) about such a binary relation which are equivalent to the existence of a $\sigma$-probability on $B$ inducing that relation.

This approach works at its best for atomless Boolean algebras (Villegas'theorem: "if a qualitative probability is atomless and monotonely continuous, then there is one and only one probability measure compatible with it, and this probability measure is countably additive"). This case is sufficient for the present purposes, since (1) a Boolean $\sigma$-algebra can be a finite-measure algebra only if the set of atoms is at most countable; (2) a atomic $\sigma$-boolean algebra with at most countably atoms is exactly $2^X$ with $X$ at most countable, and these are directly seen to be finite-measure algebras; (3) a Boolean $\sigma$-algebra with at most countably many atoms is exactly the direct product of a atomic part as above and a atomless Boolean $\sigma$-algebra.

Such a binary relation can be used in the foundations of the concept of (subjective) probability (quite analogously to the Loomis - Maeda dimension theory and its application to rings of operators), but I do not see an analogue use for conditions in Jech's style. But I understand that such a point of view is not suitable for everyone, exactly like using quantum logic to recover Hilbert space or starting geometry from synthetic axioms.

Appendix: atomless qualitative probability $\sigma$-algebras, from Villegas'paper.

[All outside square brackets is as in the paper except trivial notation editing]

[Start from a Boolean algebra of "events" with minimum $O$ and maximum $\Omega$, partial order $\subseteq$ and lattice operations $\cup$ and $\cap$. Atomless and atomic are as usual in Boolean algebras]

A relation $\leq$ between events will be called a qualitative probability, if it satisfies the following axioms:

Q1. The relation $\leq$ is a total pre-order between events, with $O$ as the first element and $\Omega$ as the last one.

[Write $\equiv$ for the equivalence induced by the pre-order $\leq$ and write $<$ for "$\leq$ but not $\equiv$"]

Q2. If $B_1\cap B_2\equiv O$, then from $A_1\leq B_1$, $A_2\leq B_2$ it follows that $A_1 \cup A_2 \leq B_1\cap B_2$ and if, in one of the first two inequalities, the sign $\leq$ is replaced by $<$, then the last one holds with the sign $\leq$ replaced by $<$.

These axioms are equivalent to those proposed by de Finetti.

[Qualitative probability $\sigma$-algebra:]

We shall say that a qualitative probability defined on a $\sigma$-algebra is monotonely continuous, if, given a monotone increasing sequence of events $A_n$ and an event $B$ such that, for every $n$, $A_n\leq B$, then $\bigcup_n A_n\leq B$.

We shall say that a probability measure $P$ is compatible with a qualitative probability $\leq$, if and only if $A\leq B$ is equivalent to $P(A)\leq P(B)$, for any pair of events $A,B$.

Theorem 3. If a qualitative probability $\sigma$-algebra is atomless, then there is one and only one compatible probability measure, and it is countably additive.

[So atomless finite-measure algebras are characterized as atomless qualitative probability $\sigma$-algebras where no nonzero event is equivalent to zero. The generic finite-measure algebras are exactly the direct product of a atomless component as above and a full power set $2^X$ with the set $X$ of atoms at most countable.]