I found the measure with density given by (with respect to Lebsegue measure) is invariant for this transformation. My question now is: how can I prove this system with this measure is ergodic? I thought to use the approach with invariant functions and Fourier series, but I'm not sure on how to write Fourier expansion with a measure different than Lebesgue's. I also thought to exploit a possible conjugacy with symbolic shift, but wasn't able to prove that and constitute a Markov partition of the unit interval. Any ideas?