4
votes
1answer
89 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
0
votes
1answer
101 views

invertible, measurable and measure preserving

$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by $T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$ and $T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$ In class we said this $T$ ...