Definition talk:Ergodic Invariant Measure

The experts I meant are e.g. the researchers who have a special interest in minor topics. I think I just said too much. The answer to your initial question is, yes, in ergodic theory we have to suppose that the measures are probability measures. It does not make sense to consider infinite measures because the measures should describe the statistical properties.--Usagiop (talk) 15:52, 8 June 2022 (UTC)

One point which I would like to see elucidated is how it makes sense that $\mu$ is called ergodic without reference to $T$. Is its ergodicity independent of the exact $T$? Is "ergodic" just sloppy for "$T$-ergodic" or "ergodic under $T$"? Disclaimer: besides hearing the word "ergodic" I am in no way familiar with the content of the theory. — Lord_Farin (talk) 19:36, 8 June 2022 (UTC)

Here, when we talk about the ergodicity, the $T$-invariant measure $\mu$ is given.

So, it is just sloppy for "This given $T$-invariant measure $\mu$ is ergodic".

You can say ergodic $T$-invariant measure but normally don't say $T$-ergodic measure--Usagiop (talk) 20:02, 8 June 2022 (UTC)