1 Introduction

Every order continuous Banach lattice with a weak unit is lattice isometric to the \(L^1\) space of a vector measure, [9, Proposition 2.4] (cf. [6, Theorem 8]). Such Banach lattices are weakly compactly generated [4, p. 193] (cf. [6, Theorem 2]) and admit an equivalent uniformly Gâteaux smooth norm, [19] (cf. [26, Theorem 2.2]). For an arbitrary Banach space X, the existence of such a norm is equivalent to being isomorphic to a subspace of a Hilbert generated Banach space, and also to \((B_{X^*},w^*)\) being uniform Eberlein compact, [13, Theorem 2] (cf. [16, Theorem 6.30]).

Typical examples of Banach lattices arising as \(L^1\) spaces of vector measures are all Banach spaces with unconditional basis, the classical spaces \(L^p(\mu )\) (for \(1\le p<\infty \) and a finite measure \(\mu \)) and Orlicz spaces over a finite measure satisfying the \(\Delta _2\)-condition. On the other hand, C[0, 1] is a separable Banach lattice which is not isomorphic (as Banach space) to the \(L^1\) space of any vector measure. In the non-separable setting, for an uncountable set \(\Gamma \) and \(1<p<\infty \), the space \(\ell ^p(\Gamma )\) is reflexive and embeds isomorphically into a Hilbert generated space, [12, Theorem 1]. As a Banach lattice, \(\ell ^p(\Gamma )\) is order continuous, but fails to have a weak unit and so it is not lattice isomorphic to the \(L^1\) space of any vector measure. In fact, \(\ell ^p(\Gamma )\) is isomorphic to the \(L^1\) space of a vector measure if and only if \(p=2\), see [26, Theorem 2.6]. Similarly, the Banach lattice \(c_0(\Gamma )\) is order continuous and Hilbert generated, but it is not isomorphic to the \(L^1\) space of any vector measure (see the Appendix).

Completely different vector measures can produce the same \(L^1\) space, see [15] for a detailed discussion. The following result was proved in [7, Theorem 1] (cf. [21, Theorem 5]):

Theorem 1.1

(G.P. Curbera) Let \(\nu \) be a vector measure defined on a \(\sigma \)-algebra \(\Sigma \) and taking values in a Banach space. If \(\nu \) is non-atomic and \(L^1(\nu )\) is separable, then there is a vector measure \(\tilde{\nu }:\Sigma \rightarrow c_0\) such that \(L^1(\nu )=L^1(\tilde{\nu })\) with equal norms.

The non-atomicity assumption in the result above cannot be dropped in general, [7, pp. 294–295]. At the conference “Integration, Vector Measures and Related Topics VI” (Bedłewo, June 2014), Z. Lipecki asked whether a non-separable version of Theorem 1.1 can be obtained by using \(c_0(\Gamma )\) as target space for large enough \(\Gamma \). Here we address this question and provide some partial answers by using certain superspaces of \(c_0(\Gamma )\). Our main results are:

Theorem 1.2

Let \(\kappa \) be an infinite cardinal. Let \(\nu \) be a vector measure defined on a \(\sigma \)-algebra \(\Sigma \) and taking values in a Banach space. If \(\nu \) is homogeneous and \(L^1(\nu )\) has density character \(\kappa \), then there is a vector measure \(\tilde{\nu }:\Sigma \rightarrow \ell ^\infty _{<}(\kappa )\) such that \(L^1(\nu )=L^1(\tilde{\nu })\) with equal norms.

Theorem 1.3

Let \(\nu \) be a vector measure defined on a \(\sigma \)-algebra \(\Sigma \) and taking values in a Banach space. If \(\nu \) is non-atomic and \(L^1(\nu )\) has density character \(\omega _1\), then there is a vector measure \(\tilde{\nu }:\Sigma \rightarrow \ell ^\infty _c(\omega _1)\) such that \(L^1(\nu )=L^1(\tilde{\nu })\) with equivalent norms.

Given an infinite cardinal \(\kappa \), we denote by \(\ell ^\infty _{<}(\kappa )\) the subspace of \(\ell ^\infty (\kappa )\) consisting of all \((x_\alpha )_{\alpha <\kappa }\in \ell ^\infty (\kappa )\) such that \(|\{\alpha<\kappa : |x_\alpha |>\varepsilon \}|<\kappa \) for every \(\varepsilon >0\). In general, \(c_0(\kappa )\) is a subspace of \(\ell ^\infty _{<}(\kappa )\). The space \(\ell ^\infty _{<}(\kappa )\) was introduced by Pełczyński and Sudakov [24] and has been studied in [3] in connection with injectivity properties of Banach spaces. For \(\kappa =\omega \) we have \(\ell ^\infty _{<}(\kappa )=c_0(\omega )\) and, therefore, Theorem 1.1 is a particular case of Theorem 1.2. If \(\kappa \) has uncountable cofinality, then \(\ell ^\infty _{<}(\kappa )\) coincides with the set of all \((x_\alpha )_{\alpha <\kappa }\in \ell ^\infty (\kappa )\) such that \(|\{\alpha<\kappa : x_\alpha \ne 0\}|<\kappa \). In particular, we have \(\ell ^\infty _{<}(\omega _1)=\ell ^\infty _{c}(\omega _1)\), the Banach space of all bounded real-valued functions on \(\omega _1\) having countable support. For information on spaces of bounded functions on an uncountable set having countable support, see [3, 17, 18, 23], [29, Section 16-1] and the references therein.

This paper is organized as follows. In Sect. 2 we introduce the basic terminology and present some preliminary results and examples of non-separable \(L^1\) spaces of vector measures. In Sect. 3 we prove our main Theorems 1.2 and 1.3. To this end we use some ideas from the alternative proof of Theorem 1.1 given in [21], together with other ingredients like Maharam’s theorem, which allows us to find a substitute for the Rademacher-type sequences used in both proofs of the separable case. We close the paper with an Appendix on linear injections into \(L^1\) spaces of vector measures.

2 Preliminaries and examples

2.1 Terminology

Our standard references are [1, 2, 8]. All our Banach spaces are real. An operator is a linear continuous map between Banach spaces. By a subspace of a Banach space we mean a closed linear subspace. The closed unit ball of a Banach space Z is denoted by \(B_Z\) and the dual of Z is denoted by \(Z^*\). By a vector measure we mean a countably additive Banach space-valued measure defined on a \(\sigma \)-algebra. The density character of a topological space T, denoted by \(\mathrm{dens}(T)\), is the minimal cardinality of a dense subset of T.

Throughout this paper \((\Omega ,\Sigma )\) is a measurable space and X is a Banach space. The set of all X-valued vector measures defined on \(\Sigma \) is denoted by \(ca(\Sigma ,X)\). The symbol \(ca_+(\Sigma )\) stands for the subset of \(ca(\Sigma ):=ca(\Sigma ,\mathbb {R})\) made up of all non-negative finite measures. The Maharam type of a non-atomic \(\mu \in ca_+(\Sigma )\) is defined as \(\mathrm{dens}(L^1(\mu ))\) and coincides with the density character of its measure algebra equipped with the Fréchet–Nikodým metric.

Let \(\nu \in ca(\Sigma ,X)\). Given \(A\in \Sigma \), we denote by \(\nu _A\) the restriction of \(\nu \) to the \(\sigma \)-algebra on A defined by \(\Sigma _A:=\{A\cap B: B\in \Sigma \}\). The composition of \(\nu \) with any \(x^*\in X^*\) is denoted by \(x^*\nu \) and belongs to \(ca(\Sigma )\). The semivariation of \(\nu \) is the function \(\Vert \nu \Vert :\Sigma \rightarrow \mathbb {R}\) defined by \(\Vert \nu \Vert (A)=\sup _{x^*\in B_{X^*}}|x^*\nu |(A)\) for all \(A\in \Sigma \) (as usual, \(|x^*\nu |\) stands for the variation of \(x^*\nu \)). Given \(\xi \in ca(\Sigma ,Y)\) (where Y is a Banach space), we write \(\nu \ll \xi \) to denote that \(\nu \) is absolutely continuous with respect to \(\xi \), meaning that \(\lim _{\Vert \xi \Vert (A)\rightarrow 0}\Vert \nu (A)\Vert =0\) or, equivalently, that \(\nu (A)=0\) whenever \(\Vert \xi \Vert (A)=0\). We say that \(\lambda \in ca_+(\Sigma )\) is a control measure of \(\nu \) if \(\lambda \ll \nu \) and \(\nu \ll \lambda \). A Rybakov control measure of \(\nu \) is a control measure of the form \(\lambda =|x^*\nu |\) for some \(x^*\in B_{X^*}\); such control measures always exist, see e.g. [8, p. 268, Theorem 2]. We say that \(\nu \) is non-atomic if some/every control measure of \(\nu \) is non-atomic in the usual sense.

A \(\Sigma \)-measurable function \(f:\Omega \rightarrow \mathbb {R}\) is said to be \(\nu \) -integrable if the following two conditions are satisfied: (i) f is \(|x^*\nu |\)-integrable for all \(x^* \in X^*\), and (ii) for each \(A \in \Sigma \), there is a vector \(\int _A f\, d\nu \in X\) such that \(x^*(\int _A f \, d\nu ) =\int _A f \, d (x^*\nu )\) for every \(x^* \in X^*\). By identifying functions which coincide \(\Vert \nu \Vert \)-a.e. we obtain the Banach lattice \(L^1(\nu )\) of all (equivalence classes of) \(\nu \)-integrable functions, equipped with the \(\Vert \nu \Vert \)-a.e. order and the norm

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}:= \sup _{x^* \in B_{X^*}} \int _\Omega |f| \, d|x^*\nu |, \quad f \in L^1(\nu ). \end{aligned}$$

Note that \(\Vert f\Vert _{L^1(\nu )}=\Vert \nu _f\Vert (\Omega )\), where \(\nu _f\in ca(\Sigma ,X)\) is defined by \(\nu _f(A):=\int _A f\, d\nu \) for all \(A\in \Sigma \). The formula

$$\begin{aligned} \Vert |f\Vert |_\nu :=\sup _{A\in \Sigma }\Big \Vert \int _A f \, d\nu \Big \Vert \end{aligned}$$

defines a norm on \(L^1(\nu )\) which is equivalent to \(\Vert \cdot \Vert _{L^1(\nu )}\), since

$$\begin{aligned} \Vert |f\Vert |_\nu \le \Vert f\Vert _{L^1(\nu )}\le 2\Vert |f\Vert |_\nu \quad \text{ for } \text{ all } \;f\in L^1(\nu ). \end{aligned}$$
(2.1)

The basic properties of the space \(L^1(\nu )\) can be found, for instance, in [22, Chapter 3]. As a Banach lattice, \(L^1(\nu )\) is order continuous and has a weak unit. We write \(\mathrm{sim} \Sigma \) to denote the set of all simple functions from \(\Omega \) to \(\mathbb {R}\), that is, linear combinations of characteristic functions \(1_A\) where \(A\in \Sigma \). Simple functions are \(\nu \)-integrable and \(\mathrm{sim} \Sigma \) is dense in \(L^1(\nu )\) (after the \(\Vert \nu \Vert \)-a.e. identification). It is easy to check that \(\mathrm{dens}(L^1(\nu ))\) coincides with the Maharam type of any control measure of \(\nu \) whenever \(\nu \) is non-atomic. We say that \(\nu \) is homogeneous if it is non-atomic and \(\mathrm{dens}(L^1(\nu ))=\mathrm{dens}(L^1(\nu _A))\) for every \(A\in \Sigma \) with \(\Vert \nu \Vert (A)>0\).

2.2 Examples

Obviously, the classical space \(L^1(\mu )\) of a finite measure \(\mu \) can be seen as the \(L^1\) space of a vector measure. The following standard construction (see e.g. [22, Corollary 3.66]) shows that the same holds for \(L^p(\mu )\) whenever \(1<p<\infty \).

Example 2.1

Let \(\mu \in ca_+(\Sigma )\) and \(1\le p < \infty \). Let \(\nu \in ca(\Sigma ,L^p(\mu ))\) be defined by \(\nu (A):=1_A\) for all \(A\in \Sigma \). Then \(L^1(\nu )=L^p(\mu )\) with equal norms.

In Example 2.6 below we will show that, for any \(1< p<\infty \) and any infinite cardinal \(\kappa \), the \(L^p\) space of the usual probability on the Cantor cube \(\{-1,1\}^\kappa \) can be realized as the \(L^1\) space of a suitable \(c_0(\kappa )\)-valued vector measure.

To this end we need some lemmas which will also be applied in Sect. 3. The first one is based on some ideas from [14, Theorem 2.1].

Lemma 2.2

Let \(\nu \in ca(\Sigma ,X)\) and let \(\Delta \) be a \(w^*\)-dense subset of \(B_{X^*}\). Then

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}=\sup _{x^*\in \Delta }\int _\Omega |f| \, d|x^*\nu | \end{aligned}$$

for every \(f\in L^1(\nu )\).

Proof

The statement is obvious for \(f=0\). Fix \(f\in L^1(\nu ){\setminus } \{0\}\) and \(\varepsilon >0\). Let \(g\in \mathrm{sim}\Sigma \) such that \(\Vert f-g\Vert _{L^1(\nu )}\le \varepsilon \) and \(g\ne 0\). Write \(g=\sum _{i=1}^p a_i 1_{A_i}\), where \(a_i\in \mathbb {R}{\setminus } \{0\}\) and the \(A_i\)’s are pairwise disjoint elements of \(\Sigma \). Choose \(x_1^*\in B_{X^*}\) such that

$$\begin{aligned} \Vert g\Vert _{L^1(\nu )} \le \int _\Omega |g| \, d|x_1^*\nu |+\varepsilon . \end{aligned}$$
(2.2)

Since \(\Delta \) is \(w^*\)-dense in \(B_{X^*}\), there is \(x_0^*\in \Delta \) such that

$$\begin{aligned} |x_1^*\nu |(A_i) \le |x_0^*\nu |(A_i)+\frac{\varepsilon }{|a_i|p} \quad \text{ for } \text{ every } \;i\in \{1,\dots ,p\}. \end{aligned}$$

Then

$$\begin{aligned} \int _\Omega |g| \, d|x_1^*\nu |= \sum _{i=1}^p|a_i||x_1^*\nu |(A_i) \le \sum _{i=1}^p|a_i||x_0^*\nu |(A_i)+\varepsilon = \int _\Omega |g| \, d|x_0^*\nu |+\varepsilon , \end{aligned}$$

which combined with (2.2) yields

$$\begin{aligned} \Vert g\Vert _{L^1(\nu )} \le \int _\Omega |g| \, d|x_0^*\nu |+2\varepsilon . \end{aligned}$$

Bearing in mind that \(\Vert f-g\Vert _{L^1(\nu )}\le \varepsilon \), we get

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}\le \Vert g\Vert _{L^1(\nu )}+\varepsilon \le \int _\Omega |g| \, d|x_0^*\nu |+3\varepsilon \le \int _\Omega |f| \, d|x_0^*\nu |+4\varepsilon . \end{aligned}$$

As \(\varepsilon >0\) is arbitrary, we have \(\Vert f\Vert _{L^1(\nu )}=\sup _{x^*\in \Delta }\int _\Omega |f| \, d|x^*\nu |\). \(\square \)

Lemma 2.3

Let \(\Gamma \) be a non-empty set and Z a subspace of \(\ell ^\infty (\Gamma )\). For each \(\gamma \in \Gamma \), denote by \(e_\gamma ^*\in B_{\ell ^\infty (\Gamma )^*}\) the \(\gamma \)-th coordinate functional. Let \(\nu \in ca(\Sigma ,Z)\). Then

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}=\sup _{\gamma \in \Gamma }\int _\Omega |f| \, d|e_\gamma ^*\nu | \end{aligned}$$

for every \(f\in L^1(\nu )\).

Proof

We denote by \(e_\gamma ^*|_Z\) the restriction of \(e_\gamma ^*\) to Z. The set \(\{e_\gamma ^*|_Z: \gamma \in \Gamma \} \subseteq B_{Z^*}\) is 1-norming and so, by the Hahn-Banach separation theorem, its absolutely convex hull \(\Delta :=\mathrm{aco}(\{e_\gamma ^*|_Z:\gamma \in \Gamma \})\) is \(w^*\)-dense in \(B_{Z^*}\). Lemma 2.2 now applies to get

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}=\sup _{e^*\in \Delta }\int _\Omega |f| \, d|e^*\nu |= \sup _{\gamma \in \Gamma }\int _\Omega |f| \, d|e_\gamma ^*\nu | \end{aligned}$$

for every \(f\in L^1(\nu )\). \(\square \)

The following lemma can be found in [21, Lemma 6].

Lemma 2.4

Let \(\nu \in ca(\Sigma ,X)\) and \(\tilde{\nu }\in ca(\Sigma ,Y)\), where Y is a Banach space. If \(\Vert f\Vert _{L^1(\nu )}=\Vert f\Vert _{L^1(\tilde{\nu })}\) for every \(f\in \mathrm{sim}\Sigma \), then \(L^1(\nu )=L^1(\tilde{\nu })\) with equal norms.

To deal with the next examples we need to introduce some terminology. Let \(\kappa \) be an infinite cardinal. For any set \(I \subseteq \kappa \) we denote by \(\rho _{I}: \{-1,1\}^\kappa \rightarrow \{-1,1\}^{I}\) the canonical projection. We say that a function \(f:\{-1,1\}^\kappa \rightarrow \mathbb {R}\) depends on coordinates from I if there is a function \(f': \{-1,1\}^{I} \rightarrow \mathbb {R}\) such that \(f=f' \circ \rho _{I}\). We say that f depends on finitely many coordinates if there is a finite set \(I \subseteq \kappa \) such that f depends on coordinates from I. Dependence on finitely many coordinates is equivalent to being a linear combination of characteristic functions of clopen subsets of \(\{-1,1\}^\kappa \). We denote by \(S(\kappa )\) the set of all real-valued functions on \(\{-1,1\}^\kappa \) depending on finitely many coordinates. We write \(\pi _\alpha :\{-1,1\}^\kappa \rightarrow \{-1,1\}\) to denote the \(\alpha \)-th coordinate projection for each \(\alpha <\kappa \).

Example 2.5

Let \(\kappa \) be an infinite cardinal, \(\lambda \) the usual probability on \(\{-1,1\}^\kappa \) and \(\Sigma \) its domain. Then

$$\begin{aligned} \nu (A):=\Big (\int _A \pi _\alpha \, d\lambda \Big )_{\alpha <\kappa }\in c_0(\kappa ) \quad \text{ for } \text{ every } \;A \in \Sigma . \end{aligned}$$

Moreover, \(\nu \in ca(\Sigma ,c_0(\kappa ))\) and \(L^1(\nu )=L^1(\lambda )\) with equal norms.

Proof

The fact that \(\nu \) takes values in \(c_0(\kappa )\) follows from the density of \(S(\kappa )\) in \(L^1(\lambda )\), cf. the proof of [25, Lemma 2.1] for more details. Clearly, \(\nu \) is finitely additive. From the inequality \(\Vert \nu (A)\Vert _{c_0(\kappa )}\le \lambda (A)\) for all \(A\in \Sigma \) it follows that \(\nu \) is countably additive. Lemma 2.3 ensures that \(\Vert f\Vert _{L^1(\nu )}=\Vert f\Vert _{L^1(\lambda )}\) for every \(f\in \mathrm{sim}\Sigma \), and then Lemma 2.4 applies to conclude that \(L^1(\nu )=L^1(\lambda )\) with equal norms. \(\square \)

The proof of the following example uses an argument which was kindly provided by G. Plebanek.

Example 2.6

Let \(\kappa \) be an infinite cardinal, \(\lambda \) the usual probability on \(\{-1,1\}^\kappa \) and \(\Sigma \) its domain. Let \(1<p<\infty \). Then there is \(\nu \in ca(\Sigma ,c_0(\kappa ))\) such that \(L^1(\nu )=L^p(\lambda )\) with equal norms.

Proof

Write \(K:=\{-1,1\}^\kappa \). Let \(1<q<\infty \) be such that \(\frac{1}{p}+\frac{1}{q}=1\) and write \(\langle f,g\rangle :=\int _K fg \, d\lambda \) for every \(f\in L^p(\lambda )\) and \(g\in L^q(\lambda )\). Since \(S(\kappa )\) is norm dense in \(L^q(\lambda )\) and \(\mathrm{dens}(L^q(\lambda ))=\kappa \), there is a set \(H \subseteq S(\kappa ) \cap B_{L^q(\lambda )}\) of cardinality \(\kappa \) which is norm dense in \(B_{L^q(\lambda )}\). Enumerate \(H=\{h_\alpha :\alpha <\kappa \}\). Each \(h_\alpha \) can be written as \(h_\alpha =h'_\alpha \circ \rho _{I_\alpha }\), where \(I_\alpha \subseteq \kappa \) is finite and \(h'_\alpha : \{-1,1\}^{I_\alpha } \rightarrow \mathbb {R}\) is a function. Since \(\kappa \) is infinite and the \(I_\alpha \)’s are finite, we can construct (inductively) an injective map \(\varphi : \kappa \rightarrow \kappa \) in such a way that \(\varphi (\alpha )\not \in I_\alpha \) for all \(\alpha <\kappa \). Define \(g_\alpha :=h_\alpha \pi _{\varphi (\alpha )}\in B_{L^q(\lambda )}\) for every \(\alpha <\kappa \).

Claim. If \((\alpha _n)\) is a sequence in \(\kappa \) with \(\alpha _n\ne \alpha _m\) whenever \(n\ne m\), then \((g_{\alpha _n})\) is weakly null in \(L^q(\lambda )\). Indeed, since \(S(\kappa )\) is norm dense in \(L^p(\lambda )=L^q(\lambda )^*\) and the sequence \((g_{\alpha _n})\) is bounded, it suffices to check that \(\langle f,g_{\alpha _n} \rangle \rightarrow 0\) as \(n\rightarrow \infty \) for every \(f\in S(\kappa )\). To this end, let us write \(f=f' \circ \rho _{I}\) for some finite set \(I \subseteq \kappa \) and some function \(f':\{-1,1\}^I \rightarrow \mathbb {R}\). Note that each \(fh_{\alpha _n}\) depends on coordinates from \(I \cup I_{\alpha _n}\). Choose \(n_0\in \mathbb {N}\) large enough such that for every \(n\ge n_0\) we have \(\varphi (\alpha _n)\not \in I\). Then for every \(n\ge n_0\) we have \(\varphi (\alpha _n)\not \in I\cup I_{\alpha _n}\) and so \(fh_{\alpha _n}\) and \(\pi _{\varphi (\alpha _n)}\) are stochastically independent, that is, \(\int _{K} fh_{\alpha _n} \pi _{\varphi (\alpha _n)} \, d\lambda =0\). Then

$$\begin{aligned} \langle f,g_{\alpha _n} \rangle =\int _{K} f g_{\alpha _n} \, d\lambda =0 \quad \text{ whenever } \;n\ge n_0. \end{aligned}$$

This proves the Claim.

From the previous claim it follows at once that for every \(A\in \Sigma \) we have

$$\begin{aligned} \nu (A):=\Big (\int _A g_\alpha \, d\lambda \Big )_{\alpha <\kappa }\in c_0(\kappa ). \end{aligned}$$

Clearly, \(\nu : \Sigma \rightarrow c_0(\kappa )\) is finitely additive and satisfies

$$\begin{aligned} \Vert \nu (A)\Vert _{c_0(\kappa )}= \sup _{\alpha<\kappa }\Big |\int _A g_\alpha \, d\lambda \Big |\le \sup _{\alpha <\kappa }\Vert 1_A\Vert _{L^p(\lambda )}\Vert g_\alpha \Vert _{L^q(\lambda )}\le \lambda (A)^{\frac{1}{p}} \quad \text{ for } \text{ all } \;A\in \Sigma , \end{aligned}$$

hence \(\nu \in ca(\Sigma ,c_0(\kappa ))\). By Lemma 2.3, the norm of any \(f\in \mathrm{sim}\Sigma \) is

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}= & {} \sup _{\alpha<\kappa }\int _{K} |f g_\alpha | \, d\lambda \\= & {} \sup _{\alpha<\kappa }\int _{K} |f h_\alpha | \, d\lambda = \sup _{\alpha <\kappa }\big \langle |f|, |h_\alpha | \big \rangle \mathop {=}\limits ^{(*)} \sup _{h\in B_{L^q(\lambda )}}\big \langle |f|, |h| \big \rangle =\Vert f\Vert _{L^p(\lambda )}, \end{aligned}$$

where equality \((*)\) follows from the norm density of \(\{h_\alpha :\alpha <\kappa \}\) in \(B_{L^q(\lambda )}\). According to Example 2.1 and Lemma 2.4, we have \(L^1(\nu )=L^p(\lambda )\) with equal norms. \(\square \)

3 Proofs of Theorems 1.2 and 1.3

In order to prove Theorems 1.2 and 1.3 we need some lemmas.

Lemma 3.1

Let \(\kappa \) be an infinite cardinal. If \(\mu \in ca_+(\Sigma )\) is homogeneous of Maharam type \(\kappa \), then there is a set \(\{\mu _{\alpha }:\alpha <\kappa \} \subseteq ca(\Sigma )\) such that:

  1. (i)

    \(|\mu _{\alpha }|=\mu \) for all \(\alpha < \kappa \);

  2. (ii)

    \((\mu _{\alpha }(E))_{\alpha <\kappa } \in c_0(\kappa )\) for all \(E\in \Sigma \).

Proof

We can suppose without loss of generality that \(\mu (\Omega )=1\). By Maharam’s theorem (see e.g. [20, p. 122, Theorem 8]), the measure algebra of \(\mu \) is isomorphic to the measure algebra of the usual probability on \(\{-1,1\}^{\kappa }\). We can now find a set \(\{g_\alpha :\alpha <\kappa \} \subseteq L^\infty (\mu )\) with \(|g_\alpha |=1\) for all \(\alpha <\kappa \) such that, for every \(E \in \Sigma \), we have \((\int _E g_\alpha \, d\mu )_{\alpha <\kappa }\in c_0(\kappa )\) (see Example 2.5). It is clear that the measures \(\mu _\alpha \in ca(\Sigma )\) defined by \(\mu _{\alpha }(E):=\int _{E}g_\alpha \, d\mu \) satisfy the required properties. \(\square \)

Lemma 3.2

Let \(\kappa \) be an infinite cardinal. Let \(\lambda \in ca_+(\Sigma )\) be homogeneous of Maharam type \(\kappa \). Let \(\{\lambda _\alpha \}_{\alpha <\kappa }\) be a family in \(ca_+(\Sigma )\) such that

$$\begin{aligned} \lambda \ll \lambda _\alpha \quad \text{ for } \text{ all } \; \alpha<\kappa \quad \text{ and } \quad \lim _{\lambda (A)\rightarrow 0} \sup _{\alpha <\kappa }\lambda _\alpha (A)=0. \end{aligned}$$

Then there is a family \(\{\mu _{\alpha }\}_{\alpha <\kappa }\) in \(ca(\Sigma )\) such that:

  1. (i)

    \(|\mu _{\alpha }|=\lambda _\alpha \) for all \(\alpha < \kappa \);

  2. (ii)

    \((\mu _{\alpha }(E))_{\alpha<\kappa } \in \ell ^\infty _{<}(\kappa )\) for all \(E\in \Sigma \).

Proof

Each \(\lambda _\alpha \) is homogeneous of Maharam type \(\kappa \), so for each \(\alpha <\kappa \) we can apply Lemma 3.1 to obtain a set \(\{\mu _{\alpha ,\beta }:\beta <\kappa \} \subseteq ca(\Sigma )\) such that:

  • \(|\mu _{\alpha ,\beta }|=\lambda _\alpha \) for all \(\beta < \kappa \);

  • \((\mu _{\alpha ,\beta }(E))_{\beta <\kappa } \in c_0(\kappa )\) for all \(E\in \Sigma \).

Fix a family \(\{A_\gamma \}_{\gamma <\kappa }\) in \(\Sigma \) such that \(\inf _{\gamma <\kappa }\lambda (E\triangle A_\gamma )=0\) for all \(E\in \Sigma \). We now distinguish two cases:

Case 1: \(\kappa =\omega \). By allowing infinitely many repetitions, we can assume further that for every \(m<\omega \) and every \(E\in \Sigma \) we have \(\inf _{n\ge m}\lambda (E\triangle A_n)=0\). For each \(n<\omega \), the set

$$\begin{aligned} B(n):=\bigcup _{m \le n} \Big \{ k<\omega : \, |\mu _{n,k}(A_m)|>\frac{1}{n+1} \Big \} \end{aligned}$$

is finite and we choose \(\beta (n)\in \omega {\setminus } B(n)\). Define \(\mu _n:=\mu _{n,\beta (n)}\in ca(\Sigma )\) for every \(n<\omega \), so that \(\{\mu _{n}\}_{n<\omega }\) satisfies (i). We next check that (ii) holds. To this end, fix \(E\in \Sigma \) and \(\varepsilon >0\). Take \(\delta >0\) such that \(\sup _{n<\omega }\lambda _n(A)\le \varepsilon \) whenever \(\lambda (A)\le \delta \). Choose \(m<\omega \) such that \(\frac{1}{m+1}\le \varepsilon \) and \(\lambda (E \triangle A_m)\le \delta \). For each \(n<\omega \) we have

$$\begin{aligned} |\mu _n(E)-\mu _n(A_m)|\le |\mu _n|(E\triangle A_m)=\lambda _n(E\triangle A_m) \le \varepsilon . \end{aligned}$$
(3.1)

Bearing in mind that \(|\mu _n(A_m)|=|\mu _{n,\beta (n)}(A_m)|\le \frac{1}{n+1} \le \varepsilon \) whenever \(n\ge m\), from (3.1) we conclude that \(|\mu _n(E)|\le 2\varepsilon \) for every \(n\ge m\). As \(\varepsilon >0\) is arbitrary, this proves that \((\mu _{n}(E))_{n<\omega } \in c_0(\omega )\). The proof of Case 1 is finished.

Case 2: \(\kappa \) is uncountable. For each \(\alpha <\kappa \), the set

$$\begin{aligned} B(\alpha ):=\bigcup _{\gamma \le \alpha } \big \{ \beta <\kappa : \, \mu _{\alpha ,\beta }(A_\gamma )\ne 0 \big \} \end{aligned}$$

has cardinality \(|B(\alpha )|<\kappa \), because \(\kappa \) is uncountable and \(\{ \beta <\kappa : \mu _{\alpha ,\beta }(A_\gamma )\ne 0 \}\) is countable for every \(\gamma <\kappa \). Then for every \(\alpha <\kappa \) we can choose \(\beta (\alpha )\in \kappa {\setminus } B(\alpha )\) and we define \(\mu _\alpha :=\mu _{\alpha ,\beta (\alpha )}\in ca(\Sigma )\). An argument similar to that of Case 1 shows that the family \(\{\mu _{\alpha }\}_{\alpha <\kappa }\) satisfies the required properties. \(\square \)

Lemma 3.3

Let \(\kappa \) be an infinite cardinal. Let \(\nu \in ca(\Sigma ,X)\) with \(\mathrm{dens}(L^1(\nu ))=\kappa \). Then there is \(C \subseteq B_{X^*}\) with \(|C|\le \kappa \) such that:

  1. (i)

    \(\nu \ll |x^*\nu |\) for all \(x^*\in C\);

  2. (ii)

    \(\Vert f\Vert _{L^1(\nu )}=\sup _{x^*\in C}\int _\Omega |f| \, d|x^*\nu |\) for all \(f\in L^1(\nu )\).

Proof

Fix a norm dense set \(\mathcal {F} \subseteq L^1(\nu )\) with \(|\mathcal {F}|=\kappa \). By the Rybakov–Walsh theorem (see e.g. [8, pp. 268–269]), the set \(\Delta :=\{x^*\in B_{X^*}: \nu \ll |x^*\nu |\}\) is norm dense (hence \(w^*\)-dense) in \(B_{X^*}\). Then for every \(f\in L^1(\nu )\) there is a countable set \(\Delta _f \subseteq \Delta \) such that

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}=\sup _{x^*\in \Delta _f}\int _\Omega |f| \, d|x^*\nu | \end{aligned}$$

(apply Lemma 2.2). It is easy to check that \(C:=\bigcup _{f\in \mathcal {F}}\Delta _f\) fulfills the required properties. \(\square \)

We arrive at the proofs of our main results:

Proof of Theorem 1.2

The Banach space in which \(\nu \) takes values is denoted by X. By Lemma 3.3 there is a collection \(\{x_\alpha ^*\}_{\alpha <\kappa }\) in \(B_{X^*}\) such that \(\nu \ll |x_\alpha ^*\nu |\) for all \(\alpha <\kappa \) and

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}=\sup _{\alpha <\kappa }\int _\Omega |f| \, d|x_\alpha ^*\nu | \quad \text{ for } \text{ all } \;f\in L^1(\nu ). \end{aligned}$$
(3.2)

Lemma 3.2 can now be applied to \(\lambda _\alpha :=|x_\alpha ^*\nu |\) and \(\lambda :=|x_0^*\nu |\) to find a family \(\{\mu _{\alpha }\}_{\alpha <\kappa }\) in \(ca(\Sigma )\) such that \(|\mu _{\alpha }|=|x_\alpha ^*\nu |\) for all \(\alpha < \kappa \) and

$$\begin{aligned} \tilde{\nu }(E):=(\mu _{\alpha }(E))_{\alpha<\kappa } \in \ell ^\infty _{<}(\kappa ) \quad \text{ for } \text{ all } \;E\in \Sigma . \end{aligned}$$

The function \(\tilde{\nu }: \Sigma \rightarrow \ell ^\infty _{<}(\kappa )\) is finitely additive. Moreover, since

$$\begin{aligned} \Vert \tilde{\nu }(E)\Vert _{\ell ^\infty _{<}(\kappa )}=\sup _{\alpha<\kappa }|\mu _\alpha (E)|\le \sup _{\alpha <\kappa }|x_\alpha ^*\nu |(E) \le \Vert \nu \Vert (E) \quad \text{ for } \text{ all } \;E\in \Sigma , \end{aligned}$$

we have \(\lim _{\lambda (E)\rightarrow 0}\Vert \tilde{\nu }(E)\Vert _{\ell ^\infty _{<}(\kappa )}=0\). It follows that \(\tilde{\nu }\in ca(\Sigma ,\ell ^\infty _{<}(\kappa ))\).

In order to prove that \(L^1(\nu )=L^1(\tilde{\nu })\) with equal norms, it suffices to check that \(\Vert f\Vert _{L^1(\nu )}=\Vert f\Vert _{L^1(\tilde{\nu })}\) for every \(f\in \mathrm{sim}\Sigma \) (Lemma 2.4). Write \(e_\alpha ^* \in B_{\ell ^\infty _{<}(\kappa )^*}\) to denote the \(\alpha \)-th coordinate projection for every \(\alpha <\kappa \). Lemma 2.3 applies to compute the norm of any \(f\in \mathrm{sim}\Sigma \) as

$$\begin{aligned} \Vert f\Vert _{L^1(\tilde{\nu })}= \sup _{\alpha<\kappa }\int _\Omega |f| \, d|e_\alpha ^*\tilde{\nu }|= \sup _{\alpha <\kappa }\int _\Omega |f| \, d|x_\alpha ^*\nu |\mathop {=}\limits ^{(3.2)}\Vert f\Vert _{L^1(\nu )}. \end{aligned}$$

The proof is complete. \(\square \)

Proof of Theorem 1.3

Let \(\mu \) be a Rybakov control measure of \(\nu \). Then \(\mu \) is non-atomic and has Maharam type \(\omega _1\). Therefore, there exist disjoint \(A,B\in \Sigma \) with \(\Omega =A\cup B\) such that \(L^1(\mu _A)\) is separable and \(\mu _B\) is homogeneous and has Maharam type \(\omega _1\) (see e.g. [20, p. 122, Theorem 7]). So, \(L^1(\nu _A)\) is separable, \(\nu _B\) is homogeneous and \(\mathrm{dens}(L^1(\nu _B))=\omega _1\). By Theorems 1.1 and 1.2 applied to \(\nu _A\) and \(\nu _B\), respectively, there exist \(\xi \in ca(\Sigma _A,c_0)\) and \(\psi \in ca(\Sigma _B,\ell ^\infty _c(\omega _1))\) such that

$$\begin{aligned} L^1(\nu _A)=L^1(\xi ) \quad \text{ and } \quad L^1(\nu _B)=L^1(\psi ) \end{aligned}$$

with equal norms. Write \(Z:=c_0\oplus _1\ell ^\infty _c(\omega _1)\) and define \(\varphi \in ca(\Sigma ,Z)\) by

$$\begin{aligned} \varphi (E):=(\xi (E\cap A),\psi (E\cap B))\quad \text{ for } \text{ all } \;E\in \Sigma . \end{aligned}$$

Fix \(f\in \mathrm{sim}\Sigma \) and denote by \(f|_A\) (resp. \(f|_B\)) its restriction to A (resp. B). Then

$$\begin{aligned} \int _E f \, d\varphi = \left( \int _{E\cap A}f|_A \, d\xi , \int _{E\cap B}f|_B \, d\psi \right) \quad \text{ for } \text{ all } \;E\in \Sigma \end{aligned}$$

and so

$$\begin{aligned} \sup _{E\in \Sigma } \Big \Vert \int _E f \, d\varphi \Big \Vert _Z = \sup _{E\in \Sigma }\Big \Vert \int _{E\cap A} f|_A \, d\xi \Big \Vert _{c_0} + \sup _{E\in \Sigma }\Big \Vert \int _{E\cap B} f|_B \, d\psi \Big \Vert _{\ell ^\infty _c(\omega _1)}. \end{aligned}$$
(3.3)

On one hand, we have

$$\begin{aligned} \Vert f\Vert _{L^1(\varphi )}&\mathop {\le }\limits ^{(2.1)} \,\, 2\sup _{E\in \Sigma } \Big \Vert \int _E f \, d\varphi \Big \Vert _Z \\&\mathop {=}\limits ^{(3.3)} \,\, 2\sup _{E\in \Sigma }\Big \Vert \int _{E\cap A} f|_A \, d\xi \Big \Vert _{c_0} + 2\sup _{E\in \Sigma }\Big \Vert \int _{E\cap B} f|_B \, d\psi \Big \Vert _{\ell ^\infty _c(\omega _1)} \\&\mathop {\le }\limits ^{(2.1)} \,\, 2 \Vert f|_A\Vert _{L^1(\xi )}+2 \Vert f|_B\Vert _{L^1(\psi )} \\&= \,\, 2 \Vert f|_A\Vert _{L^1(\nu _A)}+2 \Vert f|_B\Vert _{L^1(\nu _B)} \\&\le \,\, 4 \Vert f\Vert _{L^1(\nu )}. \end{aligned}$$

On the other hand, we also have

$$\begin{aligned} \Vert f\Vert _{L^1(\nu )}&= \,\, \Vert f1_A+f1_B\Vert _{L^1(\nu )} \\&\le \,\, \Vert f|_A\Vert _{L^1(\nu _A)}+\Vert f|_B\Vert _{L^1(\nu _B)} \\&= \,\, \Vert f|_A\Vert _{L^1(\xi )}+\Vert f|_B\Vert _{L^1(\psi )} \\&\mathop {\le }\limits ^{(2.1)} \,\, 2\sup _{E\in \Sigma } \Big \Vert \int _{E\cap A} f|_A \, d\xi \Big \Vert _{c_0}+ 2\sup _{E\in \Sigma } \Big \Vert \int _{E\cap B}f|_B \, d\psi \Big \Vert _{\ell ^\infty _c(\omega _1)} \\&\mathop {=}\limits ^{(3.3)} \,\, 2\sup _{E\in \Sigma } \Big \Vert \int _E f \, d\varphi \Big \Vert _Z \\&\mathop {\le }\limits ^{(2.1)} \,\, 2\Vert f\Vert _{L^1(\varphi )}. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{1}{4} \Vert f\Vert _{L^1(\varphi )} \le \Vert f\Vert _{L^1(\nu )}\le 2\Vert f\Vert _{L^1(\varphi )} \quad \text{ for } \text{ every } f\in \mathrm{sim}\Sigma . \end{aligned}$$

The proof of Lemma 2.4 given in [21, Lemma 6] can now be adapted straightforwardly to prove that \(L^1(\nu )=L^1(\varphi )\) with equivalent norms.

Since \(c_0\) embeds isomorphically into \(\ell ^\infty _c(\omega _1)\) and \(\ell ^\infty _c(\omega _1)\) is isomorphic to its square, the space \(Z=c_0\oplus _1\ell ^\infty _c(\omega _1)\) embeds isomorphically into \(\ell ^\infty _c(\omega _1)\). Take any isomorphic embedding \(j:Z \rightarrow \ell ^\infty _c(\omega _1)\) and define \(\tilde{\nu }:=j\circ \varphi \in ca(\Sigma ,\ell ^\infty _c(\omega _1))\). It is easy to check that \(L^1(\tilde{\nu })=L^1(\varphi )\) with equivalent norms. Then \(L^1(\nu )=L^1(\tilde{\nu })\) with equivalent norms and the proof is complete. \(\square \)