Diophantine properties of measures invariant with respect to the Gauss map

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family of measures on the real line which are Ahlfors regular and yet do not satisfy a 0-1 law for approximability.