I'm dealing with two sources regarding ergodic theory.
One is Tao's blog, which assumes $T$ an invertible measure preserving map.
The other is this book which only assumes $T$ is measure preserving. This is not a silly definition since there are interesting examples of this sort (bernouli shift).
I am getting the feeling one only needs to think about the invertible case, is this correct? What are the common methods to show this?
My attempt-
Technically we also need to deal with inverse existing a.e (i.e there is $S$ so that $ST=Id$ a.e, and $TS=Id$ a.e). But this is easily reduced to the invertible case on a subset of measure $1$.
The invertible case is very convient because of the Klusterman operator being unitary for instance, which lets the spectral theorem work when proving the mean ergodic theorem.
The book presents a general universal construction that takes a measure perserving system and presents it as a factor of one with an invertible measure perserving map. I think this allows to basically to always prove things assuming the invertible case and then use the factor to prove it in the general case.
For example in the mean ergodic case-
Suppose we $Y$ with map $T$ a factor of $X$ with map $S$. Then we have an isometric map $F: L^2 (Y)$ \to $L^2(X)$ that makes the klusterman operators commute. Then given $f \in L^2(Y)$, moving it to $L^2(X)$ the averages converge to the projection to the fixed subspace, and it's easy to pull this back to $L^2(Y)$.
Is this argument common and correct? Should I always pretend I'm living in the invertible world? If so what are other common reasons? Is there a way to use the spectral theorem directly in the noninvertible case?