$\def\bs#1{\boldsymbol#1}\def\sp{\kern.4mm}$Let $\bs K$ be either the standard real or complex topological field, and let $E$ be a Hausdorff locally convex space over $\bs K\sp$. Then saying that $E$ has the *Dunford−Pettis* property means that if we have any Banach space $F$ over $\bs K$ and any continuous linear map $u:E\to F$ such that for all bounded sets $B$ in $E$ we have $u\sp[\,B\,]$ relatively compact in $F_\sigma\,$, then for every absolutely convex compact set $K$ in $E_\sigma$ we have $u\sp[\,K\,]$ relatively compact in $F\sp$. Here $G_\sigma$ means the underlying vector space of $G$ equipped with the weak topology $\sigma(G,G')$ against the topological dual space.

If now $\mu$ is a positive measure on some set $\Omega\sp$, it is "well-known" that $L^1(\mu)$ has the Dunford−Pettis property. The standard reference book for this kind of matters is the first volume of Dunford and Schwartz' *Linear Operators* which I unfortunately do not have at hand. However, according to the information given by R. E. Edwards in his *Functional Analysis* this matter should be in Theorem VI.8.12 on page 508 that I didn't succeed to see by Google. I have a suspicion that there is the additional restriction that $\mu$ be $\sigma$−finite. Furthermore, in the references I have seen this matter considered, there is the restrictive assumption that $\mu$ be a Radon measure on some Hausdorff locally compact topological space.

Now **I am asking** whether there exist any publication where this $L^1(\mu)$ having the Dunford−Pettis property is proved for more general positive measures $\mu$ than referred to above$\sp$**?**

In particular, I am interested to know whether $L^1(\mu)$ has the Dunford−Pettis property in the case where $\mu$ is *almost decomposable* in the following sense: There are a "negligible" set $N'$ and a decomposition $\mathscr A$ of $\Omega\setminus N'$ into disjoint sets $A$ of (strictly) positive finite measure such that if we have any set $N''\subseteq\Omega$ with the property that $A\cap N''$ is included in some set $N_A$ of zero measure for all $A\in\mathscr A\sp$, then $N''$ is "negligible"; this meaning that $N''\cap B$ is included in some set, depending on $B$ and $N''$, of zero measure for all $B$ having finite measure.

**Added.** (8.12.2017) Having slept over the night, it suddenly occurred to me how the case of an *almost decomposable* positive measure is "trivially reduced" to the case of a $\sigma$-finite one. It confused me when Bill Johnson said that the "general" case could be reduced the "separable" case since there are even probability measures $\pi$ with $L^1(\pi)$ not separable. The "coin tossing" measure on $\Omega={}^{\mathbb R}\,\{\sp 0\sp, 1\sp\}$ is an example. The reduction can be done by using the "convergent sequences characterization" of the Dunford−Pettis property in the case of Banach spaces by observing that $L^1(\mu)$ is linearly homeomorphic to the $\ell^{\sp 1}\,$−sum of the family $\langle\sp L^1(\mu_A):A\in\mathscr A\sp\rangle$ when $\mu$ is almost decomposable. This solves the problem in the case of an almost decomposable positive measure.

Since at the moment I still do not have access to the book of Dunford and Schwartz, I would be grateful if in a comment someone, who has access, could confirm my suspicion that there the case of a general $\sigma$-finite positive measure is fully proved, and hence that there is no "Radon restriction". Furthermore, I am still interested to know if there possibly exists a reference where the case of positive measures that are more general than "almost decomposable" ones is considered.