Week 8 Worksheet Solutions

Home | Assessment | Notes | Worksheets | Tutorials

1. 1. Since $T$ is a bijection the sets $E$ and $T^{-1}(E)$ always have the same cardinality and therefore the same measure.
   2. Fix a set $E\subset \mathbb{Z}/p\mathbb{Z}$ with positive measure. This means there is a point $e\in E$. Since $E$ is invariant it contains the orbit of $e$. But the orbit of $e$ is all of $\mathbb{Z}/p\mathbb{Z}$ so $\mu (E)=1$.
2. As $X$ is finite we will have ergodicity if and only if all orbits are equal to all of $X$. This happens when and only when $r$ and $s$ are coprime.
3. The collection $$\{[a,b):0\le a<b\le 1\}\cup \{\varnothing \}$$ is a π-system so it suffices to check $$\lambda ([a,b))=\lambda (T^{-1}([a,b)))$$ for all $0\le a<b\le 1$. But $\lambda ([a,b))=b-a$ and $$T^{-1}([a,b))=[\frac{a}{2},\frac{b}{2})\cup [\frac{a+1}{2},\frac{b+1}{2})$$ is a union of disjoint intervals whose lengths add to $b-a$.
4. This is covered in the solutions to the Week 8 Tutorial.
5. Fix $k\in \mathbb{N}$. For each $N\in \mathbb{Z}$ define $$I_k(N)=\{x\in \mathbb{R}:{\displaystyle \frac{N}{2^k}}\le f(x)<{\displaystyle \frac{N+1}{2^k}}\}$$ and note that $T^{-1}(I_k(N))=I_k(N)$ for all $N\in \mathbb{Z}$. By ergodicity each one has either measure 1 or measure 0. As their union is all of $X$ there must be exactly one $N(k)\in \mathbb{Z}$ for which $\mu (I_k(N(k)))=1$. The sets in the decreasing sequence $$k\mapsto I_k(N(k))$$ all have full measure so their intersection - which consists of those $x\in X$ at which $f$ assumes a specific value - has measure one as well. Therefore $f$ is equal on a set of full measure to a constant function.