How to interpret a theorem stating that orbits are "uniform on average"

Question

I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf

There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following:

Here $R_{\alpha}^k (x)$ is the $k_{th}$ iteration of the rotation by $\alpha$ map: $R_\alpha(x) = x+\alpha \ (\text{mod} \ 1)$

Question:

There is a comment just above this theorem saying

The distribution of the orbit is “uniform” on average.

Does this mean that the distributions of orbits on subsets of the circle are also uniform i.e. that if we fix some set $E$ on the circle and wait for the orbit to enter $E$, it has equal probability of landing anywhere in $E$?

Answer 1

If the set $E$ has positive measure, yes.