(a) Give the definition of the Borel $\sigma$-algebra on $\mathbb{R}$ and a Borel function $f: E \rightarrow \mathbb{R}$ where $(E, \mathcal{E})$ is a measurable space.

(b) Suppose that $\left(f_{n}\right)$ is a sequence of Borel functions which converges pointwise to a function $f$. Prove that $f$ is a Borel function.

(c) Let $R_{n}:[0,1) \rightarrow \mathbb{R}$ be the function which gives the $n$th binary digit of a number in $[0,1$ ) (where we do not allow for the possibility of an infinite sequence of 1 s). Prove that $R_{n}$ is a Borel function.

(d) Let $f:[0,1)^{2} \rightarrow[0, \infty]$ be the function such that $f(x, y)$ for $x, y \in[0,1)^{2}$ is equal to the number of digits in the binary expansions of $x, y$ which disagree. Prove that $f$ is non-negative measurable.

(e) Compute the Lebesgue measure of $f^{-1}([0, \infty))$, i.e. the set of pairs of numbers in $[0,1)$ whose binary expansions disagree in a finite number of digits.