Remark

— Let X be a Lusin space (in particular, any Polish space), and f a bijective continuous mapping of X onto a (Lusin) regular space Y. One knows (TG, IX, §6, No. 7, Prop. 14) that the mapping $B \mapsto f^{-1}(B)$ is a bijection of the Borel tribe of Y onto the Borel tribe of X. The spaces X and Y are Lusin, hence strongly Radon (Prop. 3). It follows immediately that the mapping $\mu \mapsto f(\mu)$ is a bijection of the set of bounded measures on X onto the set of bounded measures on Y.(4)