  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.
