(a) Define the Borel $\sigma$-algebra $\mathcal{B}$ and the Borel functions.

(b) Give an example with proof of a set in $[0,1]$ which is not Lebesgue measurable.

(c) The Cantor set $\mathcal{C}$ is given by

$$\mathcal{C}=\left\{\sum_{k=1}^{\infty} \frac{a_{k}}{3^{k}}:\left(a_{k}\right) \text { is a sequence with } a_{k} \in\{0,2\} \text { for all } k\right\}$$

(i) Explain why $\mathcal{C}$ is Lebesgue measurable.

(ii) Compute the Lebesgue measure of $\mathcal{C}$.

(iii) Is every subset of $\mathcal{C}$ Lebesgue measurable?

(iv) Let $f:[0,1] \rightarrow \mathcal{C}$ be the function given by

$$f(x)=\sum_{k=1}^{\infty} \frac{2 a_{k}}{3^{k}} \quad \text { where } \quad a_{k}=\left\lfloor 2^{k} x\right\rfloor-2\left\lfloor 2^{k-1} x\right\rfloor$$

Explain why $f$ is a Borel function.

(v) Using the previous parts, prove the existence of a Lebesgue measurable set which is not Borel.