Geometric analysis and measure theory in general ambient spaces.

In this thesis we deal with several questions concerning measure theory and geometric analysis. In the first chapter we deal with two questions: the first one is a problem proposed by David H. Frelmin concerning Hausdorff measures on $\mathbb{R}^n$ built with a distance inducing the Euclidean topology. The second question we answer is about the absolute continuity of the pushforward measure via suitable charts in metric measure spaces. Then, we study the asymptotics as $s\to o^+$ of the $s$-fractional perimeter on Riemannian manifolds and we prove that, under a stochastic completeness assumption, various definitions of fractional Laplacian all agree. Finally, we study the regularity of harmonic maps whose domain is an open set inside an ${\rm RCD}(K,N)$ space and whose target space is a ${\rm CAT}(\kappa)$ space, proving that they are H\"older regular; moreover, assuming Lipschitz regularity, we prove a variant of the celebrated Bochner-Eells-Sampsons formula.