Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\mu)^{**}$, and we can write $L_1(\mu)^{**}=L_1(\mu)\oplus L_1(\mu)^d$, where $L_1(\mu)^d = \{x^{**}\in L_1(\mu)^{**} : |x^{**}|\wedge |x|=0 \textrm{ for each }x\in L_1(\mu)\}$.

In "Linear Operators, Part I" 1988 by Dunford and Schwarz (IV.8.16 on page 296), there is a description of $L_1(\mu)^{**}$ as the space of finitely additive bounded signed measures that are absolutely continuous with respect to $\mu$ (with the variation norm). Every such finitely additive measure that is countably additive corresponds to an element of $L_1(\mu)$ by the Radon-Nikodym theorem.

I am interested in a description of the finitely additive measures that correspond with elements of $L_1(\mu)^d$.