integral over almost sure existing derivatives

Question

Let $f$ and $g$ and $f-g$ be real-valued, Lipschitz functions, with Lipschitz constant smaller or equal to 1, on $[a,b]$ whose derivatives are positive and exist only $\lambda$-almost surely. Does then the following hold: \begin{align*} \int_r^t|\frac{d}{dx}f(x)-\frac{d}{dx}g(x)|dx \leq t-r \end{align*} I was thinking of \begin{align} \int_r^t|\frac{d}{dx}f(x)-\frac{d}{dx}g(x)|dx =\int_A\frac{d}{dx}f(x)-\frac{d}{dx}g(x)dx +\int_B\frac{d}{dx}g(x)-\frac{d}{dx}f(x)dx \end{align} with $A=\{x:\frac{d}{dx}f(x)-\frac{d}{dx}g(x)\geq 0\}$ and $B=A^c$, but I am confused about the almost sure existence of the derivatives.

Answer 1

The derivative of $f-g$ are almost surely bounded by your Lipschitz constant (the so called essential supremum). Then you can use the usual supremums norm estimate for the integral, that is, $$ \int_A f d \lambda = esssup|f| \cdot \lambda(A).$$ This holds in fact for every positive measure.