Let $X$ be a set. Recall that a Boolean algebra $\mathcal{B}$ of subsets of $X$ is a family of subsets containing the empty set, which is stable under finite union and under taking complements. As usual, let $\sigma(\mathcal{B})$ be the $\sigma$-algebra generated by $\mathcal{B}$.

(a) State the definitions of a $\sigma$-algebra, that of a measure on a measurable space, as well as the definition of a probability measure.

(b) State Carathéodory's extension theorem.

(c) Let $(X, \mathcal{F}, \mu)$ be a probability measure space. Let $\mathcal{B} \subset \mathcal{F}$ be a Boolean algebra of subsets of $X$. Let $\mathcal{C}$ be the family of all $A \in \mathcal{F}$ with the property that for every $\epsilon>0$, there is $B \in \mathcal{B}$ such that

$$\mu(A \triangle B)<\epsilon,$$

where $A \triangle B$ denotes the symmetric difference of $A$ and $B$, i.e., $A \triangle B=(A \cup B) \backslash(A \cap B)$.

(i) Show that $\sigma(\mathcal{B})$ is contained in $\mathcal{C}$. Show by example that this may fail if $\mu(X)=+\infty$.

(ii) Now assume that $(X, \mathcal{F}, \mu)=\left([0,1], \mathcal{L}_{[0,1]}, m\right)$, where $\mathcal{L}_{[0,1]}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $[0,1]$ and $m$ is the Lebesgue measure. Let $\mathcal{B}$ be the family of all finite unions of sub-intervals. Is it true that $\mathcal{C}$ is equal to $\mathcal{L}_{[0,1]}$ in this case? Justify your answer.