(a) Define the following concepts: a $\pi$-system, a $d$-system and a $\sigma$-algebra.

(b) State the Dominated Convergence Theorem.

(c) Does the set function

$$\mu(A)= \begin{cases}0 & \text { for } A \text { bounded } \\ 1 & \text { for } A \text { unbounded }\end{cases}$$

furnish an example of a Borel measure?

(d) Suppose $g:[0,1] \rightarrow[0,1]$ is a measurable function. Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0) \leqslant f(1)$. Show that the limit

$$\lim _{n \rightarrow \infty} \int_{0}^{1} f\left(g(x)^{n}\right) d x$$

exists and lies in the interval $[f(0), f(1)]$