Baire category of subset $\mathbb{R} \setminus B$ with lebesgue zero measure $B$

Question

Let $\mathbb{Q}=\{ q_i\}^{\infty}_{i=1}$.

Let $I_{ij} = \left(q_i - \delta_{ij}, q_i + \delta_{ij}\right)$, $\delta_{ij}=2^{-i-j-1}$ for all $i,j \geq 1$ be an open interval. For every $j\geq1$, let $G_{j} = \bigcup\limits_{i=1}^{\infty} I_{ij}$, and let $B = \bigcap\limits_{j\geq1}^{\infty}G_{j}$.

I want to show that $B$ is a set of Lebesgue measure zero and $\mathbb{R}\setminus B$ is first category Baire set in $\mathbb{R}$

Answer 1

You can estimate the measure of $G_j$ pretty easily. Then apply continuity of the measure for $B$.

Every $G_j$ is dense (why?) and open (why?), so the other statement follows from the fact that the complement of an open dense subset is closed and nowhere dense. So $\mathbb{R} \setminus B$ is the countable union of such sets (de Morgan).