In which spaces is every curve Lebesgue-Pettis integrable?
  In a real locally convex Hausdorff space the closed convex hull of every metrizable compact set is compact if (and only if) every continuous curve has a Pettis integral with respect to Lebesgue measure. For such spaces there is a natural concept of Bochner integrals. 
Heinrich von Weizs\"acker
math.FA