What is meant by the Borel $\sigma$-algebra on the real line $\mathbb{R}$ ?

Define the Lebesgue measure of a Borel subset of $\mathbb{R}$ using the concept of outer measure.

Let $\mu$ be the Lebesgue measure on $\mathbb{R}$. Show that, for any Borel set $B$ which is contained in the interval $[0,1]$, and for any $\varepsilon>0$, there exist $n \in \mathbb{N}$ and disjoint intervals $I_{1}, \ldots, I_{n}$ contained in $[0,1]$ such that, for $A=I_{1} \cup \cdots \cup I_{n}$, we have

$$\mu(A \triangle B) \leqslant \varepsilon$$

where $A \triangle B$ denotes the symmetric difference $(A \backslash B) \cup(B \backslash A)$.

Show that there does not exist a Borel set $B$ contained in $[0,1]$ such that, for all intervals $I$ contained in $[0,1]$,

$$\mu(B \cap I)=\mu(I) / 2$$