Let $\mu$ be a finite positive measure on a set $M$:
$$
\mu(M)<\infty.
$$
As is known, the Banach dual space $L\_\infty(\mu)^\*$ to the space $L\_\infty(\mu)$ contains $L\_1(\mu)$, but (excluding some trivial situations) does not coincide with $L\_1(\mu)$:
$$
L\_1(\mu)\subseteq L\_\infty(\mu)^\*, \qquad L\_1(\mu)\ne L\_\infty(\mu)^\*.
$$



> 
> Is there a reasonable description of $L\_1(\mu)$ as a space of functionals on $L\_\infty(\mu)$?
> 
> 
> 


For example, is the following true?



> 
> **Conjecture.** A functional $f\in L\_\infty(\mu)^\*$ belongs to $L\_1(\mu)$ if and only if 
>  $$
> f(x\_n)\underset{n\to\infty}{\longrightarrow} 0
> $$
>  for each sequence of functions $x\_n\in L\_\infty(\mu)$ which is bounded in $L\_\infty(\mu)$ as in a Banach space
>  $$
> ||x\_n||\_\infty\le 1,
> $$
>  and converges to zero [almost everywhere](http://mathworld.wolfram.com/AlmostEverywhereConvergence.html)
>  $$
> x\_n\overset{\text{almost everywhere}}{\underset{n\to\infty}{\longrightarrow}}0.
> $$
> 
> 
> 


This condition is necessary due to the [Lebesgue's dominated convergence theorem](http://mathworld.wolfram.com/LebesguesDominatedConvergenceTheorem.html). Is it sufficient?



A characterization of $L_1(\mu)$ in $L_\infty(\mu)^*$
It is shown in "Linear Operators, Part I" 1988 by Dunford and Schwarz as IV.8.16 on page 296 that the dual of $L\_\infty(\mu)$ can be identified with 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. So the problem reduces to identifying countable additivity. If $\langle A\_n\rangle$ is a disjoint sequence of measurable sets, we have that indicator functions of $\bigcup\_{n=1}^\infty A\_n\setminus\bigcup\_{n=1}^m A\_n$ converge pointwise to zero, so countable additivity follows from the given condition.


