(a) What is meant by saying that $(\Omega, \mathcal{A}, \mu)$ is a measure space? Your answer should include clear definitions of any terms used.

(b) Consider the following sequence of Borel-measurable functions on the measure space $(\mathbb{R}, \mathcal{L}, \lambda)$, with the Lebesgue $\sigma$-algebra $\mathcal{L}$ and Lebesgue measure $\lambda$ :

$$f_{n}(x)=\left\{\begin{array}{ll} 1 / n & \text { if } 0 \leqslant x \leqslant e^{n} ; \\ 0 & \text { otherwise } \end{array} \quad \text { for } n \in \mathbb{N}\right.$$

For each $p \in[1, \infty]$, decide whether the sequence $\left(f_{n}\right)_{n \in \mathbb{N}}$ converges in $L^{p}$ as $n \rightarrow \infty$.

Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge almost everywhere?

Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge in measure?

Justify your answers.

For parts (c) and (d), let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence of real-valued, Borel-measurable functions on a probability space $(\Omega, \mathcal{A}, \mu)$.

(c) Prove that $\left\{x \in \Omega: f_{n}(x)\right.$ converges to a finite limit $\} \in \mathcal{A}$.

(d) Show that $f_{n} \rightarrow 0$ almost surely if and only if $\sup _{m \geqslant n}\left|f_{m}\right| \rightarrow 0$ in probability.