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We establish a Rademacher type theorem involving Hamiltonians $H(x,p)$ undervery weak conditions in both of Euclidean and Carnot-Carath\'eodory spaces. Inparticular,$H(x,p)$ is assumed to be only measurable in the variable $x$, andto be quasiconvex and lower-semicontinuous in the variable $p$. Without thelower-semicontinuity in the variable $p$, we provide a counter example showingthe failure of such a Rademacher type theorem. Moreover, by applying such aRademacher type theorem we build up an existence result of absolute minimizersfor the corresponding $L^\infty$-functional. These improve or extend severalknown results in the literature.

A Rademacher type theorem for Hamiltonians $H(x,p)$ and application to absolute minimizers