We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.