Let the transformation decimal expansion $f : [0; 1] \rightarrow [0; 1]$, given by $f(x) = 10x − \lfloor 10x\rfloor$ and m the Lebesgue measure in $[0; 1]$.
we know that $m$ is invariant by $f$ and that $m$ is ergodic to $f$.
My question is if $f$ is uniquely ergodic, which is: only the Lebesgue measure is invariant by $f$ and ergodic. I've been trying to prove this fact but I could not. I tried to prove by observing the following: The temporal mean of $f$ is constant almost everywhere by the ergodicity and Birkhoff's theorem. To show that it is uniquely ergodic I need to show that this convergence is always. Here I got lost.
Any suggestion? or counterexample? I do not know if it really uniquely is ergodic.