Invariant measures generated by sequences of approximating transformations

Let τ : [0, 1] → [0, 1] be the map defined by τ(x) = 2x(mod 1) and let λ denote Lebesgue measure on [0, 1], which is the unique absolutely continuous τ-invariant measure. We construct sequences of transformations {Tn} 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, we prove that {μn} converges to a measure singular with respect to λ. Furthermore, we characterize this singular measure in terms of the approximating transformations. We 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.