Let $m$ be the Lebesgue measure on the real line. Recall that if $E \subseteq \mathbb{R}$ is a Borel subset, then

$$m(E)=\inf \left\{\sum_{n \geqslant 1}\left|I_{n}\right|, E \subseteq \bigcup_{n \geqslant 1} I_{n}\right\},$$

where the infimum is taken over all covers of $E$ by countably many intervals, and $|I|$ denotes the length of an interval $I$.

(a) State the definition of a Borel subset of $\mathbb{R}$.

(b) State a definition of a Lebesgue measurable subset of $\mathbb{R}$.

(c) Explain why the following sets are Borel and compute their Lebesgue measure:

$$\mathbb{Q}, \quad \mathbb{R} \backslash \mathbb{Q}, \quad \bigcap_{n \geqslant 2}\left[\frac{1}{n}, n\right] .$$

(d) State the definition of a Borel measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$.

(e) Let $f$ be a Borel measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Is it true that the subset of all $x \in \mathbb{R}$ where $f$ is continuous at $x$ is a Borel subset? Justify your answer.

(f) Let $E \subseteq[0,1]$ be a Borel subset with $m(E)=1 / 2+\alpha, \alpha>0$. Show that

$$E-E:=\{x-y: x, y \in E\}$$

contains the interval $(-2 \alpha, 2 \alpha)$.

(g) Let $E \subseteq \mathbb{R}$ be a Borel subset such that $m(E)>0$. Show that for every $\varepsilon>0$, there exists $a<b$ in $\mathbb{R}$ such that

$$m(E \cap(a, b))>(1-\varepsilon) m((a, b)) .$$

Deduce that $E-E$ contains an open interval around 0 .