Here are some definitions that might be necessary for my following question:
(I) The Quadruple $(\Omega,\mathcal{A},\mu,T)$ is called dynamical system, if $(\Omega,\mathcal{A},\mu)$ is a probability space and $ T\colon\Omega\to\Omega$ a measurable Transformation with the property $\mu\circ T^{-1}=\mu$.
(II) A set $A\in\mathcal{A}$ is called $T$-invariant, if $T^{-1}(A)=A $ a.s. (i.e. $T^{-1}(A)\Delta A$ is a null set).
(III) The system $(\Omega,\mathcal{A},\mu,T)$ is called ergodic, if for all T-invariant sets $A\in\mathcal{A}$ it is $\mu(A)\in\left\{0,1\right\}$.
Assume that $A\in\mathcal{A}$ is $T$-invarant, i.e. $\mu(T^{-1}A\Delta A)=0$. Show per induction that $\mu(T^{k}A\Delta A)=0$ for all $k\in\mathbb{Z}$.
Unfortunately, I do have some problems to do that.
The beginning is clear: For $k=-1$ it is (because $A$ is $T$-invariant $$ \mu(T^{-1}A\Delta A)=0. $$
Now assume that the statement is correct for $k\in\mathbb{Z}$. It is to show, that $$ \mu(T^{k+1}A\Delta A)=0. $$
I do not know how to do that. Can you please help me to show that?
We have for any measurable sets $A$, $B$ and $C$ that $$A\Delta C\subset (A\Delta B)\cup (B\Delta C),$$ hence for each $k\geqslant 0$, $$\mu(T^{-(k+1)}(A)\Delta A)\leqslant \mu(T^{-(k+1)}(A)\Delta T^{-k}A)+\mu(T^{-k}(A)\Delta A)=\mu(T^{-1}(A)\Delta A)+\mu(T^{-k}(A)\Delta A,$$ where the last inequality follows from the fact that $T$ preserves the measure $\mu$.