Let $(\Omega,\mathfrak{A},\mu,T)$ be a dynamic system in measure theory and $p\geq 1$. Show the implication $(1)\implies (2)$, whereat $$ (1)~~~~~\forall f\in L_{\mu}^p: f= f\circ T\implies f=\text{ const.} $$ $$ (2)~~~~~(\Omega,\mathfrak{A},\mu,T)\text{ is ergodic}. $$
I do not really know how to show that.
In another exercise I showed that $(2)$ is equivalent to $(3)$ whereat $$ (3)~~~~~\forall A\in\mathfrak{A}: T^{-1}A\subset A\implies\mu(A)\in\left\{0,1\right\}. $$ So I take a $A\in\mathfrak{A}$ with $T^{-1}A\subset A$ and now have to show that $\mu(A)\in\left\{0,1\right\}$, assuming (1)?
That (1) implies (2) follows from the definition of ergodicity (take $f$ the characteristic function of a measurable set).
For the converse, take $f$ a measurable function and define $$S(c):=\{x, f(x)\leqslant c\}.$$ The set $S(c)$ is measurable and invariant hence its measure is $0$ or $1$. Defining $c_0:=\inf\{c, \mu(S(c))=1\}$, we have that $f=c_0$ almost surely.