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

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.