We address three problems arising in the theory of infinite-dimensional dynamical systems. First, we study the extent to which the Hausdorff dimension and the dimension spectrum of a fractal measure supported on a compact subset of a Banach space are affected by a typical mapping into a finite-dimensional Euclidean space. We prove that a typical mapping preserves these quantities up to a factor involving the thickness of the support of the measure. Second, we prove a weighted Sobolev-Lieb-Thirring inequality and we use this inequality to derive a physically relevant upper bound on the dimension of the global attractor associated with the viscous lake equations. Finally, we show that in a general setting one may deduce the accuracy of the projection of a dynamical system solely from observation of the projected system.