Representation of product measure and measurability of measures of sections of sets

Let T and X be complete separable metric spaces,and T×X be their product space.Let ν be a complete Borel probability measure on T and τ a premeasure on X.From ν and τ we may define measures for the product space T×X by two different ways.In case τ is σ-finite,we show that the measures defined by these two ways are exactly the product measure ν×τ*,where τ* is the Method I measure induced by τ.Fore τ non-σ-finite,we proved under some assumptions,the functions τ(Et) and τ*(Et) are measurable on T,where ET×X,Et=｛x∈X;(t,x)∈E｝.