Something I do not understand. It is Theorem 2.1 of the article of K.R. Parthasarathy "On the category of ergodic measures, Illinois J. Math. 5 (1961), pages 648-656 Full text here: (projecteuclid link).

Does this theorem states, as a particular case, that for any self-homeomorphism of a compact metric space, the set of ergodic invariant measures is dense in the set of all invariant measures?

If not, please can someone explain me what my mistake is in trying to apply that theorem to deduce the above statement?

If yes, please can you explain how to approximate with ergodic measures the following invariant measure in the example below?

Example in the two-dimensional torus $T: (x,y) \rightarrow (x+y,y)$;
$y_1 \neq y_2$ are irrational. So $T$, if restricted to the circle $y= y_1$ and if restricted to the circle $y=y_2$, is irrational rotations. Hence the Lebesgue measures $\mu_1$ and $\mu_2$ along each of both circles, are ergodic. Take

$\mu = (\mu_1 + \mu_2)/2$. It is non-ergodic and invariant, and it seems to me that it can not be approached by ergodic measures.

denseG$_\delta$. There are of course many interesting self-homeomorphisms of compact metric spaces with any specified finite number of ergodic measures. $\endgroup$ – David Handelman Apr 14 '17 at 16:55