Suppose that fR - R is a continuous non-negative function such that ( f(x) dx = 1. Define a probability by setting P((-co, b]) = If(x) dx for all b E R, and then use Caratheodory's extension theorem to produce a Borel probability measure on R. Now prove that for any To ER, the singleton {To} is a Borel set and P(ro}) = 0