Throughout this question $(E, \mathcal{E}, \mu)$ is a measure space and $\left(f_{n}\right), f$ are measurable functions.

(a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.

(b) If $f_{n} \rightarrow f$ pointwise a.e., does $f_{n} \rightarrow f$ in measure? Give a proof or a counterexample.

(c) If $f_{n} \rightarrow f$ in measure, does $f_{n} \rightarrow f$ pointwise a.e.? Give a proof or a counterexample.

(d) Now suppose that $(E, \mathcal{E})=([0,1], \mathcal{B}([0,1]))$ and that $\mu$ is Lebesgue measure on $[0,1]$. Suppose $\left(f_{n}\right)$ is a sequence of Borel measurable functions on $[0,1]$ which converges pointwise a.e. to $f$.

(i) For each $n, k$ let $E_{n, k}=\bigcup_{m \geqslant n}\left\{x:\left|f_{m}(x)-f(x)\right|>1 / k\right\}$. Show that $\lim _{n \rightarrow \infty} \mu\left(E_{n, k}\right)=0$ for each $k \in \mathbb{N}$.

(ii) Show that for every $\epsilon>0$ there exists a set $A$ with $\mu(A)<\epsilon$ so that $f_{n} \rightarrow f$ uniformly on $[0,1] \backslash A$.

(iii) Does (ii) hold with $[0,1]$ replaced by $\mathbb{R}$ ? Give a proof or a counterexample.