I came across the following statement:
Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ with Vol}(U)<\delta: \left| \int_U f \right|<\varepsilon$$
It reminded me on uniform continuity, and since the latter shows up very frequently, I was wondering if this statement has any immediate consequences. Or maybe it's a special case of a more general theorem? How can the statement be generalized?
This can be generalized: this is a notion called uniform integrability.
It gives nice convergence properties in measure theory (in particular a generalization of the dominated convergence theorem).