Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that the distributions corresponding to these densities converge weakly to Lebesgue measure confined to the unit interval.
In other words, $$\lim_{n \to \infty}\int_{0}^{y}2^n \chi_{A_n}= y.$$ How ? I can't use any DCT/BCT here.