Model-theoretic independence
in the Banach lattices $L_{p}(\mu)$

Let $X$ be a set, $U$ a $\sigma$-algebra on $X$ and $\mu$ a measure on $U$, and let $p\in[1,\infty)$. We denote by $L_{p}(X,U,\mu)$ the space of (equivalence classes of) $U$-measurable functions $f\colon X\to\mathbb{R}$ such that $\|f\|=(\int|f|^{p}d\mu)^{1/p}<\infty$. We consider this space as a Banach lattice (complete normed vector lattice) over $\mathbb{R}$ in the usual way; in particular, the lattice operations $\wedge,\vee$ are given by pointwise maximum and minimum. In this paper we study model-theoretic stability and independence in such $L_{p}$ Banach lattices.

There are several ways to understand model-theoretic stability for classes of structures, like these, that lie outside the first order context. For the structures considered in this paper, these approaches are completely equivalent. The work of Iovino [Iov99] provides tools for understanding stability in normed space structures using the language of positive bounded formulas developed by Henson [Hen76]. (See also [HI02].) A different approach was initiated by Ben Yaacov [Ben03b] in the compact abstract theory (cat) setting [Ben03a]; first order structures and normed space structures are special cases. Roughly speaking, stability as developed in [Iov99] and [Ben03b] corresponds to the study of universal domains in which there are bounds on the size of spaces of types. Buechler and Lessmann [BL03] developed a notion of simplicity (and thus also of stability) for strongly homogeneous structures. More recently, Ben Yaacov and Usvyatsov [BU] developed local stability for metric structures in a continuous version of first order logic. (See also [BBHU08].)

In all four settings, a key ingredient is the analysis of a model-theoretic concept of independence. In [Iov99, part II, section 3] this analysis is based on a notion of non-forking, which is characterized there using definability of types. Independence is studied in [Ben03b] and in [BL03] by means of the notion of non-dividing (as defined by Shelah); in [Ben03b, section 2], non-dividing in a stable structure is also characterized via definability of types. In [BU] the local stability of continuous formulas is developed and a treatment of independence is sketched in [BBHU08]. It follows from what is proved in those papers that stability and independence are the same notions from all four points of view, for the structures to which they all apply.

In particular, for the structures studied here, these four approaches to stability and independence are equivalent. In this paper, we take the concept of non-dividing as the foundation of our study of independence.

We prove here that for each $p$ with $1\leq p<\infty$, the Banach lattices $L_{p}(\mu)$ are model-theoretically stable as normed space structures, and we give a characterization of non-dividing using concepts from analysis. See [BBHU08] for a summary of how these results can be translated into the setting of continuous first order logic for metric structures.

Krivine and Maurey [KM81] noted that $L_{p}(\mu)$ spaces are stable, in a sense that amounts to stability for quantifier-free positive formulas in the language of Banach spaces. This observation was part of a larger project, in which the main theorem is a deep subspace property of quantifier-free stable (infinite dimensional) Banach spaces: if $X$ is such a Banach space, then for some $p$ in the interval $1\leq p<\infty$, the sequence space $\ell_{p}$ embeds almost isometrically in $X$.

We strengthen this stability observation about the $L_{p}(\mu)$ spaces so that it applies to the Banach lattice setting and to arbitrary positive bounded formulas in that language (i.e., with bounded quantifiers allowed). Our general motivation is model-theoretic, in that we study independence, non-dividing, and canonical bases in these structures. In this paper we do not attempt to derive structural results that apply to all Banach lattices that are stable in the (strong) sense considered here.

Our work is organized as follows.

In section 2, we introduce basic notions from analysis and probability, such as conditional expectations and distributions.

In section 3 we recall model-theoretic results about atomless $L_{p}$ Banach lattices, concerning properties such as elimination of quantifiers [HI02] and separable categoricity [Hen76], and we review a characterization of types in terms of conditional distributions that is due to Markus Pomper [Pom00]. We prove that the Banach lattices $L_{p}(\mu)$ are $\omega$-stable (with respect to the metrics on the spaces of types that are induced by the norm). (This fact had been observed by the third author [Hen87] but not published.) All of this is done in the model theoretic context described in [HI02], which develops the language of positive bounded formulas with an approximate semantics.

In section 4 we use conditional expectations to characterize non-dividing for the theory of atomless $L_{p}$ Banach lattices, thus providing a relation between model-theoretic independence and the notion of independence used in analysis and probability.

In section 5 we give a close analysis of the space of 1-types over a given set of parameters in atomless $L_{p}$ Banach lattices. In particular, we give an explicit formula for calculating the distance metric on that space of 1-types (Corollary 5.10.) This metric is centrally important in the model theory of structures such as the ones considered here. The tools developed in this section also yield a second characterization of model-theoretic independence in these structures (Proposition 5.12.)

Finally, in section 6, we construct canonical bases for types in atomless $L_{p}$ Banach lattices using conditional slices. In particular we prove they always exist as sets of ordinary elements, i.e., without any need for imaginary sorts.

We start with a review of some results from analysis that we use throughout this paper.

Let $(X,U,\mu)$ be a measure space; that is, $X$ is a nonempty set, $U$ is a $\sigma$-algebra of subsets of $X$, and $\mu$ is a $\sigma$-additive measure on $U$ (not necessarily assumed to be finite or even $\sigma$-finite, in general).

A measure space $(X,U,\mu)$ is called decomposable (also called strictly localizable) if there exists a partition $\{X_{i}:i\in I\}\subset U$ of $X$ into measurable sets such that $\mu(X_{i})<\infty$ for all $i\in I$ and such that for any subset $A$ of $X$, $A\in U$ iff $A\cap X_{i}\in U$ for all $i\in I$ and, in that case, $\mu(A)=\sum_{i\in I}\mu(A\cap X_{i})$. When these properties hold, the partition $\{X_{i}:i\in I\}$ will be called a witness for the decomposability of $(X,U,\mu)$. (See [HLR91].)

If $(X,U,\mu)$ and $(Y,V,\nu)$ are measure spaces, we define a product measure space $(X,U,\mu)\otimes(Y,V,\nu)$ by defining $\mu\otimes\nu$ in the usual way on rectangles $A\times B$, where $A\in U$ and $B\in V$ have finite measure, and then extending to make the resulting product measure space on $X\times Y$ decomposable. When both measure spaces are $\sigma$-finite, this agrees with the usual product measure construction.

Let $1\leq p<\infty.$ A Banach lattice $E$ is an abstract $L_{p}$-space if $\|x+y\|^{p}=\|x\|^{p}+\|y\|^{p}$ whenever $x,y\in E$ and $x\wedge y=0$. Evidently $L_{p}(X,U,\mu)$ is an abstract $L_{p}$-space for every measure space $(X,U,\mu)$. For the study of $L_{p}(\mu)$ spaces, the requirement that all measure spaces be decomposable causes no loss of generality; indeed, the representation theorem for abstract $L_{p}$-spaces states that each such space is $L_{p}(X,U,\mu)$ for some decomposable measure space $(X,U,\mu)$. (Discussions of this representation theorem can be found in [LT79, pp. 15–16] and [Lac74, Chapter 5]; see [Lac74, p. 135] for the history of this result. See also the proof of Theorem 3 in [BDCK67], which was a key paper in the model theory of $L_{p}$-spaces.)

Let $E$ be any Banach lattice and $f\in E$. The positive part of $f$ is $f\vee 0$, and it is denoted $f^{+}$. The negative part of $f$ is $f^{-}=(-f)^{+}$, and one has $f=f^{+}-f^{-}$ and $|f|=f^{+}+f^{-}$. Further, $f$ is positive if $f=f^{+}$ and $f$ is negative if $-f$ is positive. For $f,g\in E$, one has $f\geq g$ iff $f-g$ is positive.

A subspace $F\subset E$ is an ideal if whenever $g\in E$ and $f\in F$ are such that $0\leq|g|\leq|f|$, one always has $g\in F$. An ideal $F$ is a band if for all collections $\{h_{j}:j\in J\}\subset F$ such that $h=\bigvee_{j\in J}h_{j}\in E$, one always has $h\in F$. By a sublattice of $E$ we mean a norm-closed linear sublattice. If $A,B$ are subsets of $E$ we write $A\leq B$ to mean that $A$ and $B$ are sublattices of $E$ and $A$ is contained in $B$.

Let $B\subset E$. One defines $B^{\perp}=\{f\in E\colon|f|\wedge|g|=0$ for all $g\in B\}$ and therefore $B^{\perp\perp}=\{f\in E\colon|f|\wedge|g|=0\text{ for all }g\in B^{\perp}\}$. It is a standard fact that $B^{\perp}$ and $B^{\perp\perp}$ are bands and $B^{\perp\perp}$ is the smallest band containing $B$. One refers to $B^{\perp}$ as the band orthogonal to $B$ and to $B^{\perp\perp}$ as the band generated by $B$. In $L_{p}$-spaces, every band is a projection band. That is, for any set $B\subset L_{p}(X,U,\mu)$, one has a lattice direct sum decomposition $L_{p}(X,U,\mu)=B^{\perp}\oplus B^{\perp\perp}$. (See [Sch74], for example.)

Let $(X,U,\mu)$ be a measure space. A measurable set $S\in U$ is an atom if $\mu(S)>0$ but there do not exist $S_{1},S_{2}\in U$ disjoint, both of positive measure, such that $S_{1}\cup S_{2}=S$. One calls $(X,U,\mu)$ atomless if it has no atoms. An atom in a Banach lattice is an element $x$ such that the ideal generated by $x$ has dimension 1. In $L_{p}(X,U,\mu)$, the atoms in the sense of this definition are exactly the elements of the form $r\chi_{S}$ where $r\neq 0$ and $S$ is an atom in the sense of measure theory. We may write $X$ as the disjoint union of two measurable sets, $X_{0}$ and $X_{1}$, such that $X_{0}$ is (up to null sets) the union of all atoms in $U$ and $X_{1}$ is atomless. Moreover, if $B$ is the set of atoms in $L_{p}(X,U,\mu)$, then $B^{\perp\perp}=L_{p}(X_{0},U_{0},\mu)$ and $B^{\perp}=L_{p}(X_{1},U_{1},\mu)$, where for each $i=0,1$, $U_{i}$ is the restriction of $U$ to $X_{i}$.

A key relationship between an abstract $L_{p}$-space $\mathcal{U}$ and a given sublattice $C$ of $\mathcal{U}$ is based on the following standard theorem. As explained in Remark 2.4, we may take $\mathcal{U}=L_{p}(X,U,\mu)$ and $C=L_{p}(Y,V,\mu)$ where $(Y,V,\mu)\subset(X,U,\mu)$.

For our characterization of model-theoretic independence, it is important that the conditional expectation operator defined above is uniquely determined by some of its Banach lattice properties. This is shown in the following result using functional analysis; a second, more elementary proof is given in Section 5. (See Proposition 5.6 and Remark 5.7).

While the definition of conditional expectation is in terms of functions on concrete measure spaces, it follows from the moreover part of Proposition 2.7 that the conditional expectation $\mathbb{E}(f|V)$ only depends on the embedding of $C$ in $\mathcal{U}$ as an abstract sublattice.

A key property of measure spaces having this “atomless over” relation is the following result [BR85, Theorem 1.3]. See also [Fre04, Lemma 331B].

The following result shows how to obtain functions with a prescribed conditional distribution. We state the result for $L_{p}$ functions; in [BR85] Berkes and Rosenthal give a proof of the corresponding result for arbitrary measurable functions.

We need an iterated version of the previous result, as stated in the next lemma. This was proved by Markus Pomper in his thesis [Pom00, Theorem 6.2.7]. Pomper gave a direct proof based on generalizing the argument of Berkes and Rosenthal to dimension $>1$. We give a different proof by iterating the $1$-dimensional result.

We use the model-theoretic tools developed in [HI02] to study normed space structures. We use the word formula to mean positive bounded formula and use the semantics of approximate satisfaction as defined there. In particular, the type of a tuple over a set, denoted $\operatorname{tp}(\bar{f},/A)$, is the collection of such formulas (with parameters from the set) approximately satisfied by the tuple. We also write $\bar{f}\equiv_{C}\bar{g}$ in order to say that $\operatorname{tp}(\bar{f}/C)=\operatorname{tp}(\bar{g}/C)$.

We study the model theory of $L_{p}$ Banach lattices in the signature $\mathcal{L}=\{0,-,(f_{q}\mid q\in\mathbb{Q}),+,\wedge,\vee,\|\ \ \|\}$, where each $f_{q}$ is interpreted as the unary function of scalar multiplication by $q$. Terms in this signature correspond to lattice polynomials; atomic formulas are of the forms $\|t\|\leq r$ and $s\leq\|t\|$ where $t$ is a term and $r,s$ are rational numbers. By $\operatorname{Th}_{\mathcal{A}}(L_{p}(X,U,\mu))$ we mean the approximate positive bounded theory of the Banach lattice $L_{p}(X,U,\mu)$ in this signature. See [HI02] for the necessary background.

Because of the direct sum decomposition of a measure space into its atomless and purely atomic parts, it is routine to extend what is done here to obtain analogous results in the general case. See the end of this section for a brief discussion.

Thus, if $\bar{f}$,$\bar{g}\in\mathcal{U}^{n}$ are two tuples and $C\subseteq\mathcal{U}$ is a set (by which we always implicitly mean that $|C|<\kappa$): $\bar{f}\equiv_{C}\bar{g}$ if and only if there exists an automorphism $\theta$ of the Banach lattice $\mathcal{U}$ which fixes $C$ (in symbols: $\theta\in\operatorname{Aut}(\mathcal{U}/C)$) sending $\bar{f}$ to $\bar{g}$.

For each cardinal $\kappa$ and each consistent set of positive bounded sentences $\Sigma$, there exists a normed space structure that is a model of $\Sigma$ and a $\kappa$-universal domain; see [HI02, Corollary 12.3 and Remark 12.4].

For the remainder of this section, $\mathcal{U}$ will be a $\kappa$-universal domain for the theory of atomless $L_{p}$ Banach lattices, where $\kappa$ is much larger than any set or collection of variables or constants under consideration. Since $\mathcal{U}$ is at least $\omega_{1}$-saturated, $\mathcal{U}$ is a metrically complete structure and there is a measure space $(X,U,\mu)$ such that $\mathcal{U}=L_{p}(X,U,\mu)$. Unless stated otherwise, sets of parameters such as $A,B,C\subset\mathcal{U}$ are required to be small.

Note that Fact 3.5 need not hold if we add constants to the language. By Fact 3.6 below, we get $(L_{p}([0,1],\mathcal{B},m),f))\equiv_{\mathcal{A}}(L_{p}([0,1],\mathcal{B},m),g))$, where $f$ and $g$ are any two norm 1, positive elements of $L_{p}([0,1],\mathcal{B},m)$. However, there are two possible isomorphism types of such structures, depending on whether or not the support of the adjoined function has measure 1 or not.

Note that Fact 3.6 fails to be true without the assumption that $\mathcal{U}$ is atomless; atoms and non-atoms can have the same quantifier-free type, but they never have the same type.

An important tool for studying non-dividing (which we do in the next section) is a characterization of types in terms of conditional distributions. The following results were proved by Markus Pomper [Pom00, Theorems 6.3.1 and 6.4.1] in his thesis. We give alternate proofs.