Part I. The Markov moment problem is to characterize the moment sequences of densities on the unit interval that are bounded by a given positive constant c. There are well-known characterizations through complex systems of non-linear inequalities. Hausdorff found simpler necessary and sufficient linear conditions. This paper gives a new proof, with some ancillary results, for example, characterizing moment sequences of bounded densities with respect to an arbitrary measure. A connection with de Finetti's theorem is then described, and moments of monotone densities are characterized. Part II. There is an abstract version of de Finetti's theorem that characterizes mixing measures with bounded or L_p densities. The general setting is reviewed; after the theorem is proved, it is specialized to coin tossing and to exponential random variables. Laplace transforms of bounded densities are characterized, simplifying a well-known theorem.