Suppose that $(\Omega,\mathcal{E},P)$ is a probability space and suppose that we have a measurable operator $T:\Omega\to\Omega$.

Recall that $T$ is said to be egodic if:

- $T$ is measure preserving: i.e. $P(T^{-1}E)=P(E)$;
- $T$ is zero-one: $E=T^{-1}(E)$ implies that $P(E)\in\{0,1\}$;

Now consider the operator $S_T:L^1(\Omega,\mathcal{E},P)\to L^1(\Omega,\mathcal{E},P)$ given by $S_T(f):=f\circ T$.

Then the Birkhoff theorem tells us that, if $T$ is ergodic, then $$ \lim_t \frac{1}{t} \sum_{j=0}^{t-1} S_T^j(f)=E[f]\quad\text{ a.s..} $$

It is known that $T$ is ergodic if and only if $S_T$ satisfies

$E[S_T(f)]=E[f]$;

$S_T(f)=f$ implies that $f$ is a constant a.s..

My question is: Is the claim in the Birkhoff theorem still true when an operator $S$ satisfies 1 and 2 above, but we do not necessarily have that $S=S_T$ for some ergodic transformation?

That is, suppose that $S_T:L^1(\Omega,\mathcal{E},P)\to L^1(\Omega,\mathcal{E},P)$ is any linear function that satisfies 1 and 2 above, is it true that $$ \lim_t \frac{1}{t} \sum_{j=0}^{t-1} S^j(f)=E[f]\quad\text{ a.s..} $$ ?

If true, can you give me some reference?