(i) Define the notion of a measurable function between measurable spaces. Show that a continuous function $\mathbb{R}^{2} \rightarrow \mathbb{R}$ is measurable with respect to the Borel $\sigma$-fields on $\mathbb{R}^{2}$ and $\mathbb{R}$.

By using this, or otherwise, show that, when $f, g: X \rightarrow \mathbb{R}$ are measurable with respect to some $\sigma$-field $\mathcal{F}$ on $X$ and the Borel $\sigma$-field on $\mathbb{R}$, then $f+g$ is also measurable.

(ii) State the Monotone Convergence Theorem for $[0, \infty]$-valued functions. Prove the Dominated Convergence Theorem.

[You may assume the Monotone Convergence Theorem but any other results about integration that you use will need to be stated carefully and proved.]

Let $X$ be the real Banach space of continuous real-valued functions on $[0,1]$ with the uniform norm. Fix $u \in X$ and define

$$T: X \rightarrow \mathbb{R} ; \quad f \mapsto \int_{0}^{1} f(t) u(t) d t$$

Show that $T$ is a bounded, linear map with norm

$$\|T\|=\int_{0}^{1}|u(t)| d t$$

Is it true, for every choice of $u$, that there is function $f \in X$ with $\|f\|=1$ and $\|T(f)\|=\|T\|$ ?