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Let T.: [0, 11 [0, 11, 0 < a < 2, denote the class of transformations defined by 2x, 0<x< It()= (2 -a) -2(1 -a)x, 2< X < 1. For a* < a <4, the transformation T. admits unique absolutely continuous invariant measures & whose density function f(x) assumes the value 0 on an interval which expands as aTl. Furthermore, the measures IL. converge weakly to the measure l1/2 = 1281/2 + {8, as aT ', where UI/2 is an invariant measure under T.,, and By denotes the Dirac measure at the point y.

Weak continuity of invariant measures for a class of piecewise monotonic transformations