Timeline for Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$

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Dec 25, 2020 at 17:25	comment	added	rpotrie		Luisin's theorem provides a (compact) set of almost full measure where the function is continuous. If the function is non constant you can also assume the restriction isn't. Then, pick density points where the function takes different values to get that.
Dec 24, 2020 at 16:11	comment	added	No One		Note that $\phi$ is measurable but not continuous. So the behavior of $\phi$ in the neighborhoods $U,V$ can be very weird... how do you make sure that " the function takes different values in at least 99% of the measure of each"?
Dec 20, 2020 at 16:44	comment	added	rpotrie		sorry, I had not seen your comment to the question that essentially points the same. I just wanted to state that Howe Moore is not really the answer to the question (though related of course, and probably every reference that makes HM) proves this too. Your comments are correct, I was not explicit enough in saying that non constant means in the sense of functions being "the same" if coincide in full measure.
Dec 20, 2020 at 16:17	comment	added	YCor		non-constant should be "not generically constant"? Also the assumption is not that it's invariant under each $g$, but for each $g$, $f-gf$ has support of measure zero.
Dec 20, 2020 at 15:40	history	answered	rpotrie	CC BY-SA 4.0