Importance of Bounded Variation Condition for Stability of an Absolutely Continuous Invariant Measure

We give an example showing that the uniformly bounded variation condition in the compactness theorem of Góra and Boyarsky [Canad. J. Math. 41 (1989), 855-869] is crucial for L1 stability of an absolutely continuous invariant measure (acim) of a piecewise expanding transformation. The same example shows that a simple method of approximating an invariant density of a piecewise expanding transformation can give the wrong results.