Homeomorphic measures in the Hilbert cube

A Borel probability measure μ in the Hubert cube is homeomorphic to the usual product measure if and only if it is positive for nonempty open sets and zero for points. The transformation can be effected by a homeomorphism equal to the identity on any prescribed μ-null Z-set. Several extension, approximation, and embedding theorems are obtained as applications. 1 * Introduction and basic theorem * Let Q = J° ° — ΐ[Tli denote the Hubert cube, where each It = [0, 1], and let λ denote Lebesgue product measure in Q restricted to Borel sets. If h is any homeo-morphism of Q onto itself, then E\-*X(h(E)) defines a Borel measure in Q which we shall denote by xh. Any measure μ that admit