= Solution
The <rational numbers> are countable, so
$$
\mathbb Q=\bigcup_{q\in\mathbb Q}\{q\}
$$
is a countable union of closed sets and hence a <Borel set>; its <Lebesgue measure> is zero. Its complement is also Borel and has infinite measure:
$$
\boxed{m(\mathbb Q)=0,\qquad m(\mathbb R\setminus\mathbb Q)=\infty.}
$$
The intervals are nested in the relevant directions, and the strongest lower and upper restrictions both occur at $n=2$:
$$
\bigcap_{n\geq2}[1/n,n]=[1/2,2].
$$
Thus this set is closed, hence Borel, and
$$
\boxed{m\left(\bigcap_{n\geq2}[1/n,n]\right)=\frac32.}
$$
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