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Showing a Transformation increases measure (Ergodic Theory)
Hoi, i want to show that the piece-wise linear map
$T: [0,1)\to[0,1)$ given by $Tx=3x$ for $x\in [0,1/3)$ and $T(x)=\frac{3}{2}x -\frac{1}{2}$ for $x\in [1/3,1)$
is Ergodic w.r.t. lebesgue measure. I know there's a lemma called Knopps Lemma that may help with this. Knopps lemma is a great help for example for showing Ergodicity in maps like $S:[0,1)\to [0,1)$ given by $Sx = 2x \mod 1$...but now we have a piecewise linear (expanding) map.
Can we for example assume that the collection $\mathcal{C}$ of fundamental intervals...($T^n(I)=[0,1)$ for any n) generated the borel-sigma algebra?
Thank you, any help is appreciated :)