Operations that preserve integrability, and truncated Riesz spaces

For any real number $p\in [1,+\infty)$, we characterise the operations $\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the operations under which, for every measure $\mu$, the set $\mathcal{L}^p(\mu)$ is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind $\sigma$-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that $\mathbb{R}$ generates this variety. From this, we exhibit a concrete model of the free Dedekind $\sigma$-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve $p$-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind $\sigma$-complete Riesz spaces with weak unit, $\mathbb{R}$ is proved to generate this variety, and a concrete model of the free Dedekind $\sigma$-complete Riesz spaces with weak unit is exhibited.


Introduction
1.1. Operations that preserve integrability. In this work we investigate the operations which are somehow implicit in the theory of integration by addressing the following question: which operations preserve integrability, in the sense that they return integrable functions when applied to integrable functions?
Let us clarify the question by recalling some definitions. For (Ω, F , µ) a measure space (with the range of µ in [0, +∞]) and p ∈ [1, +∞), we adopt the notation L p (µ) := {f : Ω → R | f is F -measurable and Ω |f | p dµ < ∞}. It is well known that, for f, g ∈ L p (µ), we have f + g ∈ L p (µ), that is, L p (µ) is closed under the pointwise addition induced by addition of real numbers + : R 2 → R. More generally, consider a set I and a function τ : R I → R, which we shall call an operation of arity |I|. We say L p (µ) is closed under τ if τ returns functions in L p (µ) when applied to functions in L p (µ), that is, for every (f i ) i∈I ⊆ L p (µ), the function τ ((f i ) i∈I ) : Ω → R given by x ∈ Ω → τ ((f i (x)) i∈I ) belongs to L p (µ). If L p (µ) is closed under τ , we also say that τ preserves p-integrability over (Ω, F , µ). Finally, we say that τ preserves p-integrability if τ preserves p-integrability over every measure space.
In Part 1 of this paper we characterise those operations that preserve integrability. Indeed, the first question we address is the following.
It turns out that, for any given p, the operations that preserve p-integrability are essentially just 0, +, ∨, λ( · ) (for each λ ∈ R), and · , in the sense that every operation that preserves p-integrability may be obtained from these by composition. This we prove in Theorem 2.3.
We also have an explicit characterisation of the operations that preserve p-integrability. Denoting with R + the set {λ ∈ R | λ 0}, for n ∈ ω and τ : R n → R, we will prove that τ preserves p-integrability precisely when τ is Borel measurable and there exist λ 0 , . . . , λ n−1 ∈ R + such that, for every x ∈ R n , we have Theorem 2.1 tackles the general case of arbitrary arity, settling Question 1.1.
In Part I we also address a variation of Question 1.1 where we restrict attention to finite measures. Recall that a measure µ on a measurable space (Ω, F ) is finite if µ(Ω) < ∞. The question becomes Question 1.2. Under which operations R I → R are L p spaces of finite measure closed? Equivalently, which operations preserve p-integrability over finite measure spaces?
As mentioned, the function constantly equal to 1 belongs to L p (µ) for every finite measure µ. We prove in Theorem 2.4 that, for any given p ∈ [1, +∞), the operations that preserve p-integrability over finite measure spaces are essentially just 0, +, ∨, λ( · ) (for each λ ∈ R), and 1, in the same sense as in the above.
Theorem 2.2 provides an explicit characterisation of the operations that preserve p-integrability over finite measure spaces. In particular, for n ∈ ω and τ : R n → R, τ preserves p-integrability over finite measure spaces precisely when τ is Borel measurable and there exist λ 0 , . . . , λ n−1 , k ∈ R + such that, for every x ∈ R n , we have 1.2. Truncated Riesz spaces and weak units. In Part 2 of this paper we investigate the equational laws satisfied by the operations that preserve p-integrability. (As it is shown by Theorems 2.1 and 2.2, the fact that an operation preserves p-integrability -over arbitrary and finite measure spaces, respectively -does not depend on the choice of p. Hence, we say that the operation preserves integrability.) We therefore work in the setting of varieties of algebras [BS81]. In this paper, under the term variety we include also infinitary varieties, i.e. varieties admitting primitive operations of infinite arity. For background please see [S l59].
We assume familiarity with the basic theory of Riesz spaces, also known as vector lattices. All needed background can be found, for example, in the standard reference [LZ71]. As usual, for a Riesz space G, we set G + := {x ∈ G | x 0}.
A truncated Riesz space is a Riesz space G endowed with a function · : G + → G + , called truncation, which satisfies the following properties for all f, g ∈ G + .
(B2) If f = 0, then f = 0. (B3) If nf = nf for every n ∈ ω, then f = 0. The notion of truncation is due to R. N. Ball [Bal14], who introduced it in the context of latticeordered groups. Please see Section 8 for further details.
Let us say that a partially ordered set B is Dedekind σ-complete if every nonempty countable subset A ⊆ B that admits an upper bound admits a supremum. Theorem 10.2 proves that the category of Dedekind σ-complete truncated Riesz spaces is a variety generated by R. This variety can be presented as having operations of finite arity only, together with the single operation of countably infinite arity. Moreover, we prove that the variety is finitely axiomatisable by equations over the theory of Riesz spaces. One consequence (Corollary 10.4) is that the free Dedekind σ-complete truncated Riesz space over a set I (exists, and) is F t (I) := f : R I → R | f preserves integrability .
We prove results analogous to the foregoing for operations that preserve integrability over finite measure spaces. An element 1 of a Riesz space G is a weak (order ) unit if 1 0 and, for all f ∈ G, f ∧1 = 0 implies f = 0. Theorem 12.2 shows that the category of Dedekind σ-complete Riesz spaces with weak unit is a variety generated by R, again with primitive operations of countable arity. It, too, is finitely axiomatisable by equations over the theory of Riesz spaces. By Corollary 12.4, the free Dedekind σ-complete Riesz space with weak unit over a set I (exists, and) is F u (I) := f : R I → R | f preserves integrability over finite measure spaces .
The varietal presentation of Dedekind σ-complete Riesz spaces with weak unit was already obtained in [Abb19]. Here we add the representation theorem for free algebras, and we establish the relationship between Dedekind σ-complete Riesz spaces with weak unit and operations that preserve integrability. The proofs in the present paper are independent of [Abb19]. On the other hand, the results in this paper do depend on a version of the Loomis-Sikorski Theorem for Riesz spaces, namely Theorem 9.3 below. A proof can be found in [BvR97], and can also be recovered from the combination of [BdPvR08] and [BvR89]. The theorem and its variants have a long history: for a fuller bibliographic account please see [BdPvR08].
1.3. Outline. In Part 1 we characterise the operations that preserve integrability, and we provide a simple set of operations that generate them. Specifically, we characterise the operations that preserve measurability, integrability, and integrability over finite measure spaces, respectively in Sections 3,4,and 5. In Section 6 we show that the operations 0, +, ∨, λ( · ) (for each λ ∈ R), and · generate the operations that preserve integrability, and that 0, +, ∨, λ( · ) (for each λ ∈ R), and 1 generate the operations that preserve integrability over finite measure spaces.
In Part 2 we prove that the categories of Dedekind σ-complete truncated Riesz spaces and Dedekind σ-complete Riesz spaces with weak unit are varieties generated by R. In more deatail, in Section 7 we define the operation , in Section 8 we define truncated lattice-ordered abelian groups, in Section 9 we prove a version of the Loomis-Sikorski Theorem for truncated ℓ-groups, in Section 10 we show the category of Dedekind σ-complete truncated Riesz spaces to be generated by R, in Section 11 we prove a version of the Loomis-Sikorski Theorem for ℓ-groups with weak unit, in Section 12 we show the category of Dedekind σ-complete Riesz spaces with weak unit to be generated by R.
Finally, as an additional result, in the Appendix we provide an explicit characterisation of the operations that preserve ∞-integrability.
Acknowledgements. The author is deeply grateful to his Ph.D. advisor prof. Vincenzo Marra for the many helpful discussions.
Part 1. Operations that preserve integrability

Main results of Part 1
In this section we state the main results of Part 1, together with the needed definitions. The first two main results (Theorems 2.1 and 2.2 below) are a characterisation of the operations that preserve p-integrability over arbitrary and finite measure spaces, respectively. The other two main results (Theorems 2.3 and 2.4) provide a set of generators for these operations. To state the theorems, we introduce a little piece of terminology.
For I set, and i ∈ I, we let π i : R I → R denote the projection onto the i-th coordinate. The cylinder σ-algebra on R I (notation: Cyl R I ) is the smallest σ-algebra which makes each projection function π i : R I → R measurable. If |I| |ω|, the cylinder σ-algebra on R I coincides with the Borel σ-algebra (see [Kal97], Lemma 1.2).
Theorem 2.1. Let I be a set, τ : R I → R and p ∈ [1, +∞). The following conditions are equivalent.
(2) τ is Cyl R I -measurable and there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J such that, for every v ∈ R I , we have Theorem 2.2. Let I be a set, τ : R I → R and p ∈ [1, +∞). The following conditions are equivalent.
(2) τ is Cyl R I -measurable and there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have Theorems 2.1 and 2.2 show that the fact that an operation preserves p-integrability -over arbitrary and finite measure spaces, respectively -does not depend on the choice of p. Hence, once Theorems 2.1 and 2.2 will be settled, we will simply say that the operation preserves integrability. The other two main results of Part I (Theorems 2.3 and 2.4 below) provide a set of generators for the operations that preserve integrability over arbitrary and finite measure spaces, respectively. To state the theorems, we start by defining, for any set C of operations τ : R Jτ → R, what we mean by operations generated by C. Given two sets Ω and I, a subset S ⊆ R Ω , and a function τ : R I → R, we say that S is closed under τ if, for every family (f i ) i∈I of elements of S, we have that τ ((f i ) i∈I ) (which is the function from Ω to R, which maps ω to τ ((f i (ω)) i∈I )) belongs to S. Consider a set C of functions τ : R Jτ → R, where the set J τ depends on τ . We say that a function f : R I → R is generated by C if f belongs to the smallest subset of R R I which contains, for each i ∈ I, the projection function π i : R I → R, and which is closed under each element of C.
Theorem 2.3. For every set I, the operations R I → R that preserve integrability are exactly those generated by the operations 0, +, ∨, λ( · ) (for each λ ∈ R), , and · .
Theorem 2.4. For every set I, the operations R I → R that preserve integrability over every finite measure space are exactly those generated by the operations 0, +, ∨, λ( · ) (for each λ ∈ R), , and 1.
The rest of Part 1 is devoted to a proof of Theorems 2.1-2.4.

Operations that preserve measurability
In this section we study measurability, which is a necessary condition for integrability. In particular, we characterise the operations that preserve measurability (Theorem 3.3). This result will be of use in the following sections as preservation of measurability is necessary to preservation of integrability (Lemma 4.2). Let us start by defining precisely what we mean by "to preserve measurability". Definition 3.1. Let τ : R I → R be a function. For (Ω, F ) a measurable space, we say that τ preserves measurability over (Ω, F ) if, for every family (f i ) i∈I of F -measurable functions from Ω to R, the function τ ((f i ) i∈I ) : Ω → R is also F -measurable. We say that τ preserves measurability if τ preserves measurability over every measurable space.
When we regard R as a measurable space, we always do so with respect to the Borel σ-algebra, denoted by B R .  Theorem 3.1.29.(ii). Now we can obtain a characterisation of the operations that preserve measurability. Theorem 3.3. Let I be a set and let τ : R I → R be a function. The following are equivalent.
For every i ∈ I, π i : R I → R is Cyl R I -measurable. Since τ preserves measurability, τ ((π i ) i∈I ) is Cyl R I -measurable. Since (π i ) i∈I : R I → R I is the identity, τ ((π i ) i∈I ) = τ •(π i ) i∈I = τ is Cyl R I -measurable.
[(3) ⇒ (1)] Let us consider a measurable space (Ω, F ) and a family (f i ) i∈I of measurable functions f i : Ω → R. Consider the function (f i ) i∈I : Ω → R I , x → (f i (x)) i∈I . We have π i • (f i ) i∈I = f i , therefore π i • (f i ) i∈I is measurable for every i ∈ I. Thus, by Lemma 3.2, (f i ) i∈I is measurable. Thus τ ((f i ) i∈I ) = τ • (f i ) i∈I is measurable, because it is a composition of measurable functions.
3.1. The operations that preserve measurability depend on countably many coordinates. A fact that will be of use in the following sections is that the operations that preserve measurability depend on countably many coordinates. This we show in Corollary 3.6 below. Let us start by recalling what is meant with "to depend on countably many coordinates". (1) Let S ⊆ R I . For J ⊆ I, we say that S depends only on J if, given any x, y ∈ R I such that x j = y j for all j ∈ J, we have x ∈ S ⇔ y ∈ S. We say that S depends on countably many coordinates if there exists a countable subset J ⊆ I such that S depends only on J.
(2) Let τ : R I → R be a function. For J ⊆ I, we say that τ depends only on J if, given any x, y ∈ R I such that x j = y j for all j ∈ J, we have τ (x) = τ (y). We say that τ depends on countably many coordinates if there exists a countable subset J ⊆ I such that τ depends only on J.
We believe that the following proposition is folklore, but we were not able to locate an appropriate reference.
Proposition 3.5. If τ : R I → R is Cyl R I -measurable, then τ depends on countably many coordinates.
Proof. First, every element of Cyl R I depends on countably many coordinates: indeed, the set of elements of Cyl R I which depend on countably many coordinates is a σ-subalgebra of Cyl R I which makes the projection functions measurable (see also 254M(c) in [Fre01]). Second, let τ : R I → R be Cyl R I -measurable. The idea that we will use is that τ is determined by the family (τ −1 ((a, +∞))) a∈Q . For every a ∈ Q, there exists a countable subset J ⊆ I such that the measurable set τ −1 ((a, +∞)) depends only on J a . Then J := a∈Q J a has the property that, for each b ∈ Q, τ −1 ((b, +∞)) depends only on J. We claim that τ depends only on J. Let x, y ∈ R I be such that x j = y j for every j ∈ J. We shall prove τ (x) = τ (y). Suppose τ (x) = τ (y). Without loss of generality, τ (x) < τ (y). Let a ∈ Q be such that τ (x) < a < τ (y). Then x / ∈ τ −1 ((a, +∞)) and y ∈ τ −1 ((a, +∞)). This implies that it is not true that τ −1 ((a, +∞)) depends only on J.
Corollary 3.6. Let I be a set and τ : R I → R be a function. If τ preserves measurability, then τ depends on countably many coordinates.
Proof. If τ preserves measurability, then τ is Cyl R I -measurable by Theorem 3.3. By Proposition 3.5, τ depends on countably many coordinates.
3.2. The case of uncountable Polish spaces. The remaining results in this section are not used in the proofs of our main results.
One may think that, for an operation τ : R I → R, the condition "τ preserve measurability over every measurable space" is too strong because we may not be interested in all measurable spaces. However, Proposition 3.7 shows that this condition is equivalent to "τ preserve measurability over (R, B R )" (if τ has countable arity).
Proposition 3.7. For a set I such that |I| |ω| and a function τ : R I → R, τ preserves measurability if, and only if, τ preserves measurability on (R, B R ).
Proof. If I = ∅, then τ is a constant function. Hence τ preserves measurability over every measurable space. Let us consider the case I = ∅. By Theorem 3.3, τ preserves measurability if, and only if, τ preserves measurability over (R, Cyl R I ). Since R I and R are uncountable Polish spaces with Borel σ-algebras Cyl R I and B R respectively, (R I , Cyl R I ) and (R, B R ) are isomorphic measurable spaces (see [Sri98], Theorem 3.3.13). (Recall that an isomorphism of measurable spaces (Ω, F ) and (Ω ′ , F ′ ) is a bijective measurable function f : Ω → Ω ′ such that its inverse is measurable.) Remark 3.8. In Proposition 3.7 above, one may replace the measurable space (R, B R ) by any of its isomorphic copies. In particular, one may replace it with the measurable space given by any uncountable Polish space endowed with its Borel σ-algebra (see [Sri98], Chapter 3).

Operations that preserve integrability
The goal of this section is to prove Theorem 2.1, i.e. to characterise the operations that preserve p-integrability.
An immediate consequence of Remark 4.1 is the following lemma.
Lemma 4.3. Let (Ω, F , µ) be a measure space, and let f, g : Ω → R be functions, and let λ ∈ R. Then the following properties hold.
Lemma 4.4. Let (Ω, F , µ) be a measure space, I a set, τ : R I → R an operation that preserves measurability over (Ω, F ) and p ∈ [1, +∞). If there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J such that, for every v ∈ R I , we have |τ (v)| j∈J λ j |v j |, then τ preserves p-integrability over (Ω, F , µ).
Proof. Let (f i ) i∈I be a family in L p (µ); since τ preserves measurability over (Ω, This shows that the condition |τ (v)| j∈J λ j |v j | is sufficient for preservation of p-integrability. We are left to prove the converse direction: when τ does not satisfy this condition, there exists a measure space over which τ does not preserve p-integrability. As we shall see, at least when the arity of τ is countable, this space can always be taken to be (R, B R , Leb) where Leb is the restriction to B R of the Lebesgue measure, and this happens because (R, B R , Leb) is what we call a partitionable measure space.
Definition 4.5. A measure space (Ω, F , µ) is called partitionable if, for every sequence (a n ) n∈ω of elements of R + , there exists a sequence (A n ) n∈ω of disjoint elements of F such that µ(A n ) = a n .
The role of partitionable measure spaces is clarified by the following result.
Lemma 4.7. Let (Ω, F , µ) be a measure space, let p ∈ [1, +∞), let I be a set and let τ : R I → R be a function. Suppose |I| |ω| and suppose (Ω, F , µ) is partitionable. If τ preserves p-integrability over (Ω, F , µ), then there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J such that, for every v ∈ R I , we have Proof. We give the proof for I = ω. The case |I| < |ω| relies on an analogous argument.
We suppose, contrapositively, that, for every finite subset of indices J ⊆ I and every J-tuple (λ j ) j∈J of nonnegative real numbers, there exists v ∈ R I such that |τ (v)| > j∈J λ j |v j |; we shall prove that τ does not preserve p-integrability. For each n ∈ ω, we let v n be an element of (1) The following chain of inequalities holds. (2) The first inequality holds for some M ∈ R + because with the condition n > i we ignore finitely many terms of the series, while with the condition v n i = 0 we ignore some null terms. The third inequality holds because n > i ⇒ i ∈ {0, . . . , n − 1}.
From eqs.
Lemma 4.8. If I is a set, τ : R I → R a function, p ∈ [1, +∞) and (Ω, F , µ) a partitionable measure space, then the following conditions are equivalent.

Examples.
Example 4.9. Let n ∈ ω and τ : R n → R. Then τ preserves p-integrability if, and only if, τ is Borel measurable and there exist λ 0 , . . . , λ n−1 ∈ R + such that, for every x ∈ R n , we have Example 4.10. A function τ : R ω → R preserves p-integrability if, and only if, τ is Borel measurable and there exist a finite subset of indices J ⊆ ω and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have

4.2.
The case of (R, B R , Leb) and the discrete case. The remaining results in this section are not used in the proofs of our main results. One may think that, for an operation τ : R I → R, the condition "τ preserve p-integrability over every measure space" is too strong because we may not be interested in all measure spaces. However, Proposition 4.11 shows that this condition is equivalent to "τ preserve p-integrability over (R, B R , Leb)" (if τ has countable arity), and Proposition 4.13 provides an analogous result for a discrete measure space.
We next provide an analogue of Proposition 4.11 for a discrete measure space. We denote by P(X) the power set of a set X.

Operations that preserve integrability over finite measure spaces
The goal of this section is to prove Theorem 2.2, i.e. to characterise the operations that preserve p-integrability over finite measure spaces. We follow the same strategy of Section 4, with the appropriate adjustments.
Lemma 5.2. Let (Ω, F , µ) be a finite measure space, I a set, τ : R I → R an operation that preserves measurability over (Ω, F ) and p ∈ [1, +∞). If there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have |τ (v)| k + j∈J λ j |v j |, then τ preserves p-integrability over (Ω, F , µ).
Proof. Let (f i ) i∈I be a family in L p (µ); since τ preserves measurability over (Ω, F ), we have that It is not difficult to see that no finite measure space is partitionable: thus we replace the concept of partitionability with a slightly different one.
Definition 5.3. A measure space (Ω, F , µ) is called conditionally partitionable if there exists a sequence (b n ) n∈ω of strictly positive real numbers such that, for every sequence (a n ) n∈ω of elements of R + satisfying a n b n for every n ∈ ω, there exists a sequence (A n ) n∈ω of disjoint elements of F such that µ(A n ) = a n .
where Leb is the Lebesgue measure, is conditionally partitionable (take b n = 1 2 n+1 ). Lemma 5.5. Let (Ω, F , µ) be a measure space, let p ∈ [1, +∞), let I be a set and let τ : R I → R be a function. Suppose that |I| |ω| and that (Ω, F , µ) is conditionally partitionable. If τ preserves p-integrability over (Ω, F , µ), then there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have Proof. We give the proof for I = ω. The case |I| < |ω| relies on an analogous argument.
We suppose, contrapositively, that, for every finite subset of indices J ⊆ I, every J-tuple (λ j ) j∈J of nonnegative real numbers and every k ∈ R + , there exists v ∈ R I such that |τ (v)| > k + j∈J λ j |v j |; we shall prove that τ does not preserve p-integrability. Since (Ω, F , µ) is conditionally partitionable, there exists a sequence (b n ) n∈ω of strictly positive real numbers such that, for every sequence (a n ) n∈ω of elements of R + satisfying a n b n for every n ∈ ω, there exists a sequence (A n ) n∈ω of disjoint elements of F such that µ(A n ) = a n .
For each n ∈ ω, we let v n be an element of , the remaining part of the proof runs as for Lemma 4.8.
(3) τ is Cyl R I -measurable and there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have |τ (v)| k + j∈J λ j |v j |. Proof.

Examples.
Example 5.7. Let n ∈ ω and τ : R n → R. Then τ preserves p-integrability over every finite measure space if, and only if, τ is Borel measurable and there exist λ 0 , . . . , λ n−1 , k ∈ R + such that, for every x ∈ R n , we have Example 5.8. A function τ : R ω → R preserves p-integrability over every finite measure space if, and only if, τ is Borel measurable and there exist a finite subset of indices J ⊆ ω and nonnegative real numbers (λ j ) j∈J and k such that, for every v ∈ R I , we have

5.2.
The case of ([0, 1], B [0,1] , Leb) and the discrete case. The remaining results in this section are not used in the proofs of our main results.
One may think that, for an operation τ : R I → R, the condition "τ preserve p-integrability over every finite measure space" is too strong because we may not be interested in all finite measure spaces. However, Proposition 5.9 shows that this condition is equivalent to "τ preserve p-integrability over ([0, 1], B [0,1] , Leb)" (at least when τ has countable arity), and Proposition 5.11 provides an analogous result for a discrete finite measure space. Similarly to the case of arbitrary measure, we next provide an analogue of Proposition 5.9 for a discrete finite measure space.
Moreover, for any pair of distinct n, m ∈ ω, the sets A n and A m are disjoint. This proves that (X, P(X), ν) is conditionally partitionable.
The set X is countably infinite, hence (X, P(X)) and (ω, P(ω)) are isomorphic measurable spaces, which concludes the proof.

Generation
The goal of this section is to prove Theorems 2.3 and 2.4, which exhibit a generating set for the class of operations that preserve integrability over arbitrary and finite measure spaces, respectively.
As it is shown by Theorems 2.1 and 2.2, the fact that an operation preserves p-integrabilityover arbitrary and finite measure spaces, respectively -does not depend on the choice of p. Hence, we say that the operation preserves integrability.
From the operations 0, +, ∨ and λ( · ) (for each λ ∈ R) we generate the operations Additionally, using , we generate Let Ω be a set and let S ⊆ R Ω . We let σ(S) denote the smallest σ-algebra F of subsets of Ω such that every s ∈ S is F -measurable.
Lemma 6.2. Let Ω be a set, let A ⊆ P(Ω), let K be an element of the σ-algebra of subsets of Ω generated by A, and let K ⊆ Y ⊆ Ω. Then K belongs to any σ-algebra G of subsets of Y such that A ∩ Y ∈ G for each A ∈ A.
Given S ⊆ R Ω , we denote by S the closure of S under 0, +, ∨, λ( · ) (for each λ ∈ R), and · . Given A ⊆ Ω, we write ½ A for the characteristic function of A in Ω.
Lemma 6.6. Let S ⊆ R Ω , let g ∈ S and let f ∈ R Ω be σ(S)-measurable and such that |f | g. Then f ∈ S .
Proof. First, we prove the statement for f 0. Given that f is positive and σ(S)-measurable, f is the supremum in R Ω of a positive increasing sequence (s n ) n∈ω of σ(S)-measurable simple functions (see [Rud87], Theorem 1.17). By Lemma 6.5, s n ∈ S for every n ∈ ω. Hence For f not necessarily positive, the previous part of the proof shows that f + and f − belong to S .
Lemma 6.7. Let (Ω, F ) be a measurable space, and, for each n ∈ ω, let f n : Ω → R be a measurable function. If, for every x ∈ Ω, sup n∈ω f n (x) ∈ R, then sup f n : Ω → R is measurable. Analogously, if, for every x ∈ Ω, inf n∈ω f n (x) ∈ R, then the function inf n∈ω f n : Ω → R is measurable.
It is worth recalling that, in the proof of Theorem 2.3, the role of the truncation operation · lies in Lemma 6.4.
Proof of Theorem 2.4. Note that the operations 0, +, ∨, λ( · ) (for each λ ∈ R), and 1 preserve integrability over finite measure spaces. Moreover, by definition, the class of the operations that preserve integrability over finite measure spaces is closed under every integrability-preserving operation and contains the projection functions. Therefore, every operation generated by 0, +, ∨, λ( · ) (for each λ ∈ R), and 1 preserves integrability over every finite measure space.

The operation
We now investigate the operation , defined on R in Section 6, for more general lattices. Given a Dedekind σ-complete (not necessarily bounded) lattice B we write for the operation on B of countably infinite arity defined as (g, f 0 , f 1 , . . . ) := sup n∈ω {f n ∧ g}.
Proposition 7.1. If B is a Dedekind σ-complete lattice, the following properties hold for every g, h ∈ B, (f n ) n∈ω ⊆ B.
Conversely, we have the following.
Proposition 7.2. If B is a lattice endowed with an operation of countably infinite arity which satisfies (TS1), (TS2) and (TS3), then B is Dedekind σ-complete and g n∈ω f n = sup n∈ω {f n ∧ g}.
Proof. By induction on k ∈ ω, (TS2) entails Suppose now that f n ∧ g h for every n ∈ ω. Then This shows g n∈ω f n = sup n∈ω {f n ∧ g}. To prove that B is Dedekind σ-complete, let (f n ) n∈ω ⊆ B and g ∈ B be such that f n g for all n ∈ ω. Then A map between two partially ordered sets is σ-continuous if it preserves all existing countable suprema.
Proposition 7.3. Let ϕ : B → C be a lattice morphism between two Dedekind σ-complete lattices. Then ϕ is σ-continuous if, and only if, ϕ preserves .
For the converse implication, suppose that ϕ is σ-continuous. Let (f n ) n∈ω ⊆ B and g ∈ B. Then Hence, ϕ preserves .
Remark 7.4. Propositions 7.1, 7.2 and 7.3 show that, whenever V is a variety with a lattice reduct, then its subcategory of Dedekind σ-complete objects, with σ-continuous morphisms, is a variety which has, as primitive operations, the operations of V together with , and, as axioms, the axioms of V together with (TS1), (TS2) and (TS3).

Truncated ℓ-groups
We assume familiarity with the basic theory of ℓ-groups. All needed background can be found, for example, in the standard reference [BKW77]. In [Bal14], R. N. Ball defines a truncated ℓ-group as an abelian divisible ℓ-group that is endowed with a function · : G + → G + , called truncation, which has the following properties for all f, g ∈ G + .
In this paper, we do not assume divisibility. The truncation · may be extended to an operation on G, by setting f = f + − f − . Here, as is standard, we set f + := f ∨ 0, and f − := −(f ∧ 0). Then, Ball's definition may be reformulated as follows.
Definition 8.1. A truncated ℓ-group is an abelian ℓ-group that is endowed with a unary operation · : G → G, called truncation, which has the following properties.
Note that (T1), (T2) and (T3) are (essentially) equational axioms. This is evident for (T1); (T2) can be written as ∀f f + ∧0 = 0; (T3) is the conjunction of the two equations ∀f, g f + ∧g + ∨f + = f + and ∀f f + ∨ f + = f + . The axioms (T4) and (T5) cannot be expressed in such equational terms. However, as we shall see, this becomes possible when we add the hypothesis of Dedekind σcompleteness.
It is well-known that a Dedekind σ-complete ℓ-group is archimedean and thus abelian. Let G be a Dedekind σ-complete ℓ-group, endowed with a unary operation · . We denote by (T4 ′ ) and (T5 ′ ) the following properties, which may or may not hold in G.
We shall use the following standard distributivity result.
We denote by σℓG t the category whose objects are Dedekind σ-complete truncated ℓ-groups, and whose morphisms are σ-continuous ℓ-homomorphisms that preserve · . Since Axioms (T1), (T2), (T3), (T4 ′ ) and (T5 ′ ) are equational, σℓG t is a variety, whose operations are the operations of ℓ-groups, together with · and , and whose axioms are the axioms of ℓ-groups, together with the following ones.
9. The Loomis-Sikorski Theorem for truncated ℓ-groups Definition 9.1. Given a set X, a σ-ideal of subsets of X is a set I of subsets of X such that the following conditions hold.
If I is a σ-ideal of subsets of X, we say that a property P holds for I-almost every x ∈ X if {x ∈ X | P does not hold for x} ∈ I. A σ-ideal I of subsets of X induces on R X an equivalence relation ∼, defined by f ∼ g if, and only if, f (x) = g(x) for I-almost every x ∈ X. We write R X I for the quotient R X ∼ . Every operation τ of countable arity on R induces an operationτ on R X I , by The assumption that I is closed under countable unions guarantees that this definition is well posed. Therefore, by Remark 7.4, R X I is a Dedekind σ-complete truncated ℓ-group.
The aim of this section is to prove the following theorem.
Theorem 9.2 (Loomis-Sikorski Theorem for truncated ℓ-groups). Let G be a Dedekind σ-complete truncated ℓ-group. Then there exist a set X, a σ-ideal I of subsets of X and an injective σcontinuous ℓ-homomorphism ι : G ֒→ R X I such that, for every f ∈ G, ι f = ι(f ) ∧ [1] I . We will give a proof that is rather self-contained, with the main exception of the use of Theorem 9.3 below. Anyway, we believe that a shorter (but not self-contained) way to prove Theorem 9.2 above (even in the less restrictive hypothesis that G is an archimedean truncated ℓ-group) may be the following. First, show that the divisible hull G d of G admits a truncation that extends the truncation of G. Then, embed G d in R X I via Theorem 5.3.6.
(1) in [Bal14]. Finally, using arguments similar to those in Theorem 6.2 in [Mun99], show that this embedding preserves all countable suprema.
Theorem 9.3 (Loomis-Sikorski Theorem for Riesz spaces). Let G be a Dedekind σ-complete Riesz space. Then there exist a set X, a σ-ideal I of subsets of X and an injective σ-continuous Riesz morphism ι : G ֒→ R X I . For a proof of Theorem 9.3 see [BvR97], or [BdPvR08] and [BvR89].
Our strategy to prove Theorem 9.2 is the following. Lemma 9.12 will prove Theorem 9.2 for countably generated algebras. This will imply that R generates the variety of Dedekind σ-complete truncated ℓ-groups, and from this fact Theorem 9.2 is derived.
(1) g n∈ω f n = sup n∈ω {f n ∧ g}, and therefore f 0 ∧ g g n∈ω f n g. Hence, Since T is a convex ℓ-subgroup of G, (1) and (2) imply that T is closed under and · .
Lemma 9.6. Let X be a set, and I a σ-ideal of subsets of X. Let (g n ) n∈ω be a sequence of functions from X to R. Suppose that, for I-almost every x ∈ X, sup n∈ω g n (x) ∈ R. Then the set {[g n ] I | n ∈ ω} admits a supremum in R X I .
Proof. Let A ∈ I be such that, for every x ∈ X \ A, sup n∈ω g n (x) ∈ R. Let v : X → R be any function such that, for every x ∈ X \ A, v(x) = sup n∈ω g n (x). Then [v] I is the supremum of {[g n ] I | n ∈ ω} in R X I .
Lemma 9.7. Let G be a Dedekind σ-complete truncated ℓ-group, let f ∈ G + and let (f i ) i∈ω ⊆ G + . Then We prove the opposite inequality. By (T3), for every k ∈ ω, we have f k ∧ (if ) if , and therefore we have Lemma 9.8. Let G be an abelian ℓ-group, let a ∈ G and let u ∈ G + . Then, (a + ∧ u) − a − = a ∧ u.
Lemma 9.9. Let G be a countably generated Dedekind σ-complete truncated ℓ-group. Then there exist a set X, a σ-ideal I of subsets of X, an injective σ-continuous ℓ-homomorphism ι : G ֒→ R X I and an element u ∈ R X I such that, for every f ∈ G, Proof. By Corollary 9.4, there exist a set X, a σ-ideal I of subsets of X and an injective σcontinuous ℓ-homomorphism ι : G ֒→ R X I . Let S be a countable generating set of G and let F := {|s 0 | + · · · + |s n−1 | | s 0 , . . . , s n−1 ∈ S}. Let us enumerate F as F = {f 0 , f 1 , f 2 , . . . }. We shall prove that the set ι f n | n ∈ ω , admits a supremum u ∈ R X I that satisfies the statement of the lemma. By Lemma 9.7, for each n ∈ ω, we have f n = fn i∈ω if n − ifn k∈ω f k . Since ι is a σ-continuous ℓ-homomorphism, using Proposition 7.3, we have the following.
(1 ′ ) For each n ∈ ω, g n (x) = gn(x) i∈ω . Let x be such that (1 ′ ) hold. Suppose by way of contradiction that sup n∈ω g n (x) = ∞. Then there exists n ∈ ω such that g n (x) > 0. Therefore, we have g n (x) = k∈ω g k (x) = ig n (x), a contradiction. Therefore, sup n∈ω g n (x) ∈ R holds for each x ∈ X satisfying (1 ′ ), and thus for I-almost every x ∈ X. By Lemma 9.6, the set {[g n ] I | n ∈ ω} = ι f n | n ∈ ω admits a supremum u.
Let f ∈ G + . Then, Let G be a Dedekind σ-complete ℓ-group, let H ⊆ G, and let u ∈ G. We say that u is a weak unit for H if u 0 and, for every h ∈ H, |h| = |h| n∈ω n(|h| ∧ u).
Remark 9.10. We will see in Lemma 11.2 that a weak unit for G in the sense above is the same as a weak unit of G in the usual sense.
Lemma 9.11. Let Y be a set, J a σ-ideal of subsets of Y , H ⊆ R Y J an ℓ-subgroup, and u ∈ R Y J a weak unit for H. Then, there exists a set X, a σ-ideal I of subsets of X, and a σ-continuous ℓ-homomorphism ψ : R Y J → R X I such that the restriction of ψ to H is injective and ψ(u) J for the natural quotient map, and similarly for [ · ] I : R X ։ R X I . Since ker [ · ] J ⊆ ker [ · ] I • ( · ) |X , by the universal property of the quotient there exists a unique σ-continuous ℓ-homomorphism ρ : R Y J → R X I such that the following diagram commutes.
For every λ ∈ R + \ {0}, the function λ( · ) : R → R which maps x to λx is an isomorphism of Dedekind σ-complete ℓ-groups. Indeed, its inverse is the map 1 λ ( · ). Then, the map m : R X → R X which maps f to the function m(f ) defined by (m(f ))(x) = 1 v(x) f (x) is an isomorphism of Dedekind σ-complete ℓ-groups; indeed, its inverse is m −1 : R X → R X defined by (m −1 (g))(x) = v(x)g(x). Therefore, there exists an isomorphism η : R X I ∼ − → R X I of Dedekind σ-complete ℓ-groups which makes the following diagram commute.
We have the following commutative diagram.

R generates Dedekind σ-complete truncated Riesz spaces
Theorem 10.1 (Loomis-Sikorski for truncated Riesz spaces). Let G be a Dedekind σ-complete truncated Riesz space. Then there exist a set X, a σ-ideal I of subsets of X, and an injective σ-continuous Riesz morphism ι : G ֒→ R X I such that, for every f ∈ G, ι f = ι(f ) ∧ [1] I . Proof. By Theorem 9.2, there exist a set X, a σ-ideal I of subsets of X, and an injective σcontinuous ℓ-homomorphism ι : G ֒→ R X I such that, for every f ∈ G, ι f = ι(f ) ∧ [1] I . Since R X I is Dedekind σ-complete, it is archimedean; by Corollary 11.53 in [Sch97], ι is a Riesz morphism.
We can now obtain the first main result of Part 2, as a consequence of Theorem 10.1.
Theorem 10.2. The variety σRS t of Dedekind σ-complete truncated Riesz spaces is generated by R.
Proof. Let G be a Dedekind σ-complete truncated Riesz space. By Theorem 10.1, there exist a set X, a σ-ideal I of subsets of X, and an injective σ-continuous Riesz morphism ι : G ֒→ R X I such that, for every f ∈ G, ι f = ι(f ) ∧ [1] I . Regarding R X I as an object of σRS t with the structure induced from R, we conclude that G is a subalgebra of a quotient of a power of R.
Remark 10.3. From Theorem 7.4 in [Abb19], it follows that R actually generates σRS t as a quasivariety, where quasi-equations are allowed to have countably many premises only.
Corollary 10.4. For any set I, is the Dedekind σ-complete truncated Riesz space freely generated by the projections π i : R I → R (i ∈ I).
Proof. By Theorem 10.2, the variety σRS t of Dedekind σ-complete truncated Riesz spaces is generated by R. Therefore, by a standard result in general algebra, the smallest σRS t -subalgebra S of R R I that contains the set of projection functions {π i : R I → R | i ∈ I} is freely generated by the projection functions. The set S is the smallest subset of R R I that contains, for each i ∈ I, the projection function π i : R I → R, and which is closed under every primitive operation of σRS t . By Theorem 2.4, S consists precisely of all operations R I → R that preserve integrability. An application of Theorem 2.1 completes the proof.
Write π : I → F t (I) for the function π(i) = π i . Then, Corollary 10.4 asserts the following. For any set I, for every Dedekind σ-complete truncated Riesz space G, for every function f : I → G, there exists a unique σ-continuous truncation-preserving Riesz morphism ϕ : F t (I) → G such that the following diagram commutes.
11. The Loomis-Sikorski Theorem for ℓ-groups with weak unit An element 1 of an abelian ℓ-group G is a weak unit if 1 0 and, for every f ∈ G, f ∧ 1 = 0 implies f = 0.
Remark 11.1. Let G be an archimedean abelian ℓ-group, and let 1 be a weak unit. Then f → f ∧ 1 is a truncation. Indeed, the following show that (T1-T5) hold.
Lemma 11.2. Let G be a Dedekind σ-complete ℓ-group G, and let 1 ∈ G. Then, 1 is a weak unit if, and only if, the following conditions hold.
Note that, in the language of Dedekind σ-complete ℓ-groups, axioms (W1) and (W2) are equational. Indeed, (W1) corresponds to 1 ∧ 0 = 0, and (W2) corresponds to ∀f f + = f + n∈ω n(f + ∧ 1). Theorem 11.3 (Loomis-Sikorski Theorem for ℓ-groups with weak unit). Let G be a Dedekind σ-complete ℓ-group with weak unit 1. Then there exist a set X, a σ-ideal I of subsets of X, and an injective σ-continuous ℓ-homomorphism ι : G ֒→ R X I such that ι(1) = [1] I . Proof. By Remark 11.1, G is a Dedekind σ-complete truncated ℓ-group, with the truncation given by f → f ∧1. Then, by Theorem 9.2, there exist a set Y , a σ-ideal J of subsets of Y and an injective σ-continuous ℓ-homomorphism ϕ : G ֒→ R X I such that, for every f ∈ G, ϕ(f ∧ 1) = ϕ(f ) ∧ [1] J . The element ϕ(1) is a weak unit for the image of G under ϕ. Therefore, by Lemma 9.11, there exists a set X, a σ-ideal I of subsets of X, and a σ-continuous ℓ-homomorphism ψ : R Y J → R X I such that the restriction of ψ to H is injective and ψ(ϕ(1)) = [1] I . The function ι := ψ • ϕ has the desired properties.
Corollary 11.4. The variety of Dedekind σ-complete ℓ-groups with weak unit is generated by R.
Proof. Let G be a Dedekind σ-complete ℓ-group with weak unit. By Theorem 11.3, G is a subalgebra of a quotient of a power of R.
12. R generates Dedekind σ-complete Riesz spaces with weak unit Theorem 12.1 (Loomis-Sikorski for Riesz spaces with weak unit). Let G be a Dedekind σ-complete Riesz space with weak unit. Then there exist a set X, a σ-ideal I of subsets of X, and an injective σ-continuous Riesz morphism ι : G ֒→ R X I such that ι(1) = [1] I . Proof. By Theorem 10.1, there exist a set X, a σ-ideal I of subsets of X and an injective σcontinuous ℓ-homomorphism ι : G ֒→ R X I such that, for every f ∈ G, ι(1) = [1] I . Since R X I is Dedekind σ-complete, and thus archimedean, by Corollary 11.53 in [Sch97], ι is a Riesz morphism.
As the second main result of Part 2, we now deduce a theorem that was already obtained in [Abb19].
Theorem 12.2. The variety σRS u of Dedekind σ-complete Riesz spaces with weak unit is generated by R.
Proof. Let G be a Dedekind σ-complete truncated Riesz space. By Theorem 12.1, G is a subalgebra of a quotient of a power of R.
Remark 12.3. It was shown in [Abb19] that R actually generates σRS u as a quasi-variety, in the sense of Remark 10.3.
Corollary 12.4. For any set I, F u (I) := {f : R I → R | f is Cyl R I -measurable and ∃J ⊆ I finite, ∃(λ j ) j∈J ⊆ R + , ∃k ∈ R + : |f | k + j∈J λ j |π j |} = = {f : R I → R | f preserves integrability over finite measure spaces} is the Dedekind σ-complete Riesz space with weak unit freely generated by the elements {π i } i∈I , where, for i ∈ I, π i : R I → R is the projection on the i-th coordinate.
The proof is analogous to the proof of Corollary 10.4, and F u (I) is characterised by a universal property analogous to the one that characterises F t (I).
Appendix A. Operations that preserve ∞-integrability In Section 4 it has been shown that, for any p ∈ [1, +∞), a function τ : R I → R preserves pintegrability if, and only if, τ is Cyl R I -measurable and there exist a finite subset of indices J ⊆ I and nonnegative real numbers (λ j ) j∈J such that, for every v ∈ R I , we have |τ (v)| j∈J λ j |v j |. Does the same hold for p = ∞? The answer is no. Indeed, the function ( · ) 2 : R → R, x → x 2 is an example of operation which preserves ∞-integrability but not p-integrability, for every p ∈ [1, +∞). In Theorem A.5, we will answer the following question.
Question A.1. Which operations R I → R preserve ∞-integrability?
We will see that an operation R I → R preserve ∞-integrability if, and only if, roughly speaking, it is measurable and it maps coordinatewise-bounded subsets of R I onto bounded subsets of R. To make this precise, we introduce some definitions.
Given a measure space (Ω, F , µ), we define L ∞ (µ) as the set of F -measurable functions from Ω to R that are bounded outside of a measurable set of null µ-measure.
We can now state the answer to Question A.1 precisely. Let I be a set and let τ : R I → R be a function. Then τ preserves ∞-integrability if, and only if, τ is Cyl R I -measurable and, for every (M i ) i∈I ⊆ R + , the restriction of τ to i∈I [−M i , M i ] is bounded. This will follow from Theorem A.5.
A.1. Operations that preserve boundedness. As a preliminary step, in Theorem A.4, we characterise the operations which preserve boundedness. Definition A.3. Let I be a set, τ : R I → R. We say that τ preserves boundedness if for every set Ω and every family (f i ) i∈I of bounded functions f i : Ω → R, we have that τ ((f i ) i∈I ) : Ω → R is also bounded.
Theorem A.4. Let I be a set and τ : R I → R. The following conditions are equivalent.
(2) For every (M i ) i∈I ⊆ R + , the restriction of τ to i∈I [−M i , M i ] is bounded.