Let $(\Omega,\mathfrak{A},\mu,T)$ be a dynamic system in measure theory and $p\geq 1$. Show that the following statements are equivalent: $$ (1)~~~~~(\Omega,\mathfrak{A},\mu,T)\text{ is ergodic}\\(2)~~~~~\text{For any measurable function }f\colon\Omega\to\mathbb{R}\text{ it is: }f=f\circ T\text{ a.s }\implies f=\text{const. a.s.}\\(3)~~~~~\forall f\in L_{\mu}^p: f=f\circ T\implies f=\text{const.} $$
Here Ergodic system, show an implication I have shown $(3)\implies (1)$. Here Show: Ergodic implies $f=\text{const a.s.}$ I tried $(1)\implies (2)$ and $(2)\implies (1)$.
Now what remains is to show that $(1)\implies (3)$. And I have some problems to do so.
Consider a function $f\in L_{\mu}^p$ with $f=f\circ T$.
Now I have to show, that $f=\text{const}$:
How can I do that (using (1))?
I only can show that $f$ is constant almost surely: because $f$ is measurable and therefore from (1) it follows by (2) and the fact that $f=f\circ T$ even surely (i.e. especially almost surely) that $f$ is constant a.s.
But I do not manage it to show that $f$ is constant surely...
Can you please help me?
In general, we may not have $f=f\circ T$ almost everywhere. Take indeed $N$ a non-empty invariant set of zero measure and replace $f$ by $g:=f\chi_N$, which satisfies $g=g\circ T$. Then when $f=c$ a.e. with $c\neq 0$, the function $g$ is not constant.