A characterization of non-atomic probabilities on [0,1] with nowhere dense supports (Q913968)

For a Borel probability measure \(\mu\) on [0,1] with \(F_{\mu}\) its cumulative distribution function let \(T_{\mu}\) be the countable collection of open intervals \(\{I_ j:\;j\in N\}\) in [0,1] on which \(F_{\mu}\) is constant. \(\mu\) is non-atomic with nowhere dense support iff defining \(F_{\mu}(I_ j)=Y_ j\) for \(j\in N\) yields a dense subset \(\{Y_ j:\;j\in N\}\) of [0,1]. This result extends to finitely additive \(\mu\). It is shown that all cumulative distribution functions are of the form \(F_{\mu}\) with \(\mu\) non-atomic and giving probability 1 to rationals in [0,1].