I've been wondering about the dual of $L^1(\mu)$ when $\mu$ is not $\sigma$-finite.
Just got a copy of Cohn Measure Theory. I see various results showing the dual is $L^\infty$ for some non-$\sigma$-finite measures, but of course that's simply not true for a general measure; so far I don't see anything about what the dual actually is in general. I'm wondering:
Say $g$ is locally meaasurable if $g\chi_E$ is measurable for every measurable set $E$ with $\mu(E)<\infty$ (equivalently, if $fg$ is measurable for every $f\in L^1$).
Question. Suppose $\mu$ is a measure and $\Lambda\in L^1(\mu)^*$. Does it follow that there exists a bounded locally measurable function $g$ with $\Lambda f = \int fg$?
Probably it's a standard counterexample and I'm just being ignorant again. But the few counterexamples to $(L^1)^*=L^\infty$ that I know are not counterexamples to this:
Example. Say $X$ is an uncountable set, $A$ is the algebra of countable or co-countable sets, and $\mu$ is counting measure on $X$, restricted to $A$. It's easy to see that if $\Lambda\in (L^1)^*$ then there exists a bounded function $g$ with $\Lambda f=\int fg$, and here every function is locally measurable. (If $g$ is bounded and not mesasurable then $f\mapsto\int fg$ is an element of $(L^1)^*$ that is not induced by an $L^\infty$ function.)
Warning: If like me you find yourself skimming through Cohn without reading the first few chapters carefully, Proposition 3.3.5 will seem clearly false:
Cohn 3.3.5, paraphrased. If $\mu$ is a measure and $g\in L^\infty(\mu)$ then $||g||_\infty=||g||_{(L^1)^*}$.
$\newcommand\set[1]{\{#1\}}$ Apparent Counterexample. Let $X=\set{1,2}$, $\mu(\set 0)=1$, $\mu(\set 1)=\infty$. Let $g=\chi_{\set 1}$. Then $||g||_\infty=1$, $||g||_{(L^1)^*}=0$.
Resolution. Cohn's $||g||_\infty$ is not what you think. Say the measurable set $N$ is locally null if $\mu(N\cap E)=0$ for every measurable set $E$ with $\mu(E)<\infty$. He defines $||g||_\infty$ to be the infimum of the $M$ such that $\set{x:|f(x)|\ge M}$ is locally null. (Of course in the $\sigma$-finite case that's equivalent to what seems to me to be the more standard definition.)
(If anyone familiar with the book pointed out other similar sources of possible Cohnfusion that would be appreciated...)
