Consider the function $f(x):[0,1]\to [0,1]$ given by

$\{\begin{array}{ll}2x& 0\le x\le \frac{1}{2}\\ x-\frac{1}{2}& \frac{1}{2}<x\le 1\end{array}$

I found the measure with density given by $\rho =\frac{4}{3}{\chi}_{[0,\frac{1}{2}]}+\frac{2}{3}{\chi}_{(\frac{1}{2},1]}$ (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 $[0,\frac{1}{2}]$ and $(\frac{1}{2},1]$ constitute a Markov partition of the unit interval. Any ideas?

$\{\begin{array}{ll}2x& 0\le x\le \frac{1}{2}\\ x-\frac{1}{2}& \frac{1}{2}<x\le 1\end{array}$

I found the measure with density given by $\rho =\frac{4}{3}{\chi}_{[0,\frac{1}{2}]}+\frac{2}{3}{\chi}_{(\frac{1}{2},1]}$ (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 $[0,\frac{1}{2}]$ and $(\frac{1}{2},1]$ constitute a Markov partition of the unit interval. Any ideas?