We present a novel approach to measure concentration that works through a deeper investigation of the semigroup method. In particular, we show how couplings and rates of convergence of Markov chains can be used to obtain concentration bounds. As an application, we obtain a measure concentration result for random unitary matrices and other kinds of Haar-distributed random variables, which allows...

‎for a given measure space $(x,{mathscr b},mu)$ we construct all measure spaces $(y,{mathscr c},lambda)$ in which $(x,{mathscr b},mu)$ is embeddable‎. ‎the construction is modeled on the ultrafilter construction of the stone--v{c}ech compactification of a completely regular topological space‎. ‎under certain conditions the construction simplifies‎. ‎examples are given when this simplification o...