This is a comment which I hope will be of interest to you, but it will be too long for this format. The type of conditions that you mention were analysed in some detail by Walter Schachermayer in the 70´s in the context of functional analysis. I am not sure if the term measure space has a universally accepted meaning but the basic setting he used was that of a positive Radon measure on a locally compact space which is bounded on compacta. One can also work with a $\sigma$-finite measure on a $\sigma$-algebra but beyond this one risks treading on "Here be dragons" territory.
The general setting is the scale of $L^p$ spaces. As is well known, the boundary case $p=\infty$ presents a number of disadvantages (bad density and duality properties, too rigorous a notion of convergence for many purposes, problems with tensor products and exponential type laws,...). It shares these with many spaces of bounded objects (continuous functions on a completely regular space, holomorphic functions, operator algebras, in particular von Neumann algebras, ...). This problem has been addressed by many mathematicians, in many contexts, but the basic idea goes back to Saks in his proof of what is now known as the Vitali-Hahn-Saks theorem. In our context, it consists of replacing the norm with the finest locally convex topology $\beta_1$ which agrees with that of $L^1_{\text{loc}}$ (the local $L^1$-topology) on the unit ball. This construction occurs frequently in the literature, under such monikers as "strict topology", "mixed topology",....
Then Schachermayer´s result is the equivalence of the following conditions on a subset $B$ of §L^1$;
- a) for every $\epsilon>0$ there is a $\delta>0$ so that if $A$ is an integrable subset of the space with $\mu(A)\leq \delta$, then $$ \int_A |x(t)|d\mu\leq \epsilon $$ for each $x$ in $B$;
1b) for each $\epsilon >0$, there is a compact subset $K$ so that for each $x$ in $B$, $\int |x(t) |d\mu\leq \epsilon$, the integral being over the complement of $A$.
$B$ is $\beta_1$ equicontinuous;
$B$ is relatively weakly compact.
This means that $\beta_1$ is the Mackey topology.
These results are written up in the third chapter of the monograph "Saks Spaces and Applications to Functional Analysis".