We have provided a common "uniform principle" behind several types of weak
convergences for a sequence of probability measures on a locally compact
separable metric space. Included are the weak*, the weak, the setwise convergences
as well as the convergence for the weak topologies induced by the Riemann
integrable functions and the bounded semi-continous functions. This unifom
principle also holds for some weak convergences in Kothe spaces. The Banach
lattice property of the underlying spaces is crucial.
We have also provided some results concerning the setwise convergence
of a sequence of measures and Fatou and Lebesgue convergence theorems for
sequences of measures and mixe sequences of functions and measures.