Let τ:[0,1]⇒[0,1] be the map defined by τ(x)=2x(mod1) and let λ denote Lebesgue measure on [0,1], which is the unique absolutely continuous τ-invariant measure. The authors construct sequences of transformations {τn} such that τn⇒τ uniformly as n⇒•, but the sequence {μn} of associated absolutely continuous τn-invariant measures does not converge to λ, not even weakly. Indeed, they prove that {μn} converges to a measure singular with respect to λ. Furthermore, the authors characterize this singular measure in terms of the approximating transformations. They also show that any τ-ergodic invariant measure can be realized as the weak limit of a sequence of absolutely continuous invariant measures associated with appropriate approximating transformations.
