Lipschitz numbers and fine differentiability on metric measure spaces

On a metric space equipped with a doubling measure supporting a Poincaré inequality, we introduce capacity-relaxed infinitesimal Lipschitz numbers and characterize Newtonian functions in $N^{1,p}$, $1\le p\le \mathrm{\infty }$, and functions of bounded variation. If $X$ is additionally $Q$-regular, then every BV function is $1$-finely differentiable at $\mu$-almost every point, and its fine differential agrees with its approximate Cheeger differential. We also extend a Sobolev regularity criterion from continuous to $p$-quasicontinuous functions.