Density point (${\lim }_{r\downarrow 0}|A\cap B(x,r)|/|B(x,r)|=1$)

A point at which a measurable set occupies an asymptotically full fraction of small balls, as expressed above with Lebesgue measure. The Lebesgue density theorem says that almost every point of a measurable set is a density point. On the real line, the same limiting conclusion holds for arbitrary shrinking intervals containing that point, since each interval lies in a centered interval of at most twice its length.