Bounded moment criterion for weak convergence

For random variables supported in one fixed compact interval, convergence of every moment is equivalent to weak convergence. One direction tests monomials; the other follows by uniformly approximating continuous functions by polynomials.

Two probability measures supported on one compact interval are equal if all their moments are equal. Indeed, equality on monomials gives equality on polynomials, the Weierstrass approximation theorem extends it to every continuous function on the interval, and compactly supported continuous functions determine a finite Borel measure.