Let $(X, {\cal \tau}, \mu)$ be a probability space and $F$ some vector subspace of $L^1(X)$. I can look at the set $$\hat{F} = \{f \in L^1 \mid \forall \varepsilon>0, \ \exists \,g,h \in F \hbox{ such that } g \leq f \leq h \hbox{ and } \int h-g < \varepsilon\}$$

Is $\hat{F}$ the closure or the completion of $F$ for some topology? If so, which one? Does it come from a distance or a norm?

Note that this is not the completion with respect to the $L^1$ norm in general. Such extension occurs for example in Riemann integration, in the proof of the Weyl criterion for equidistribution or in the portmanteau theorem in probability.