Short question concerning derivation of Measure for non continuous CDF

by saguru   Last Updated October 10, 2019 03:20 AM - source

This might be totally trivial, but it puzzles me nevertheless:

Let $F(x)$ be the CDF of $x \sim U[0,1]$. Let $\Delta(x)= \{ \begin{matrix} 0 \quad x<1/3 \\ 1/3 \quad 1/3\leq x <2/3 \\ 2/3 \quad 2/3 \leq x <1 \\ 1 \quad 1 \leq x \end{matrix}$.

Define

$G(x) = \int_0^{min(x,\Delta(x))} f(x) dx = F(min(x,\Delta(x))$

If G was continuous we could just take derivatives to get the density g. Since G is not continuous, I struggle to derive the measure dG as a function of f or F. Is there anybody who could point me in the right direction?