Let $\mu$ be the Lebesgue measure, and $+$ be addition modulo $1$ in the interval $[0,1)$.


**Question1:** Is there a closed set $C\subset [0,1)$ of positive measure such that for any countable set $D\subset [0,1)$, we have $\mu(C+D)<1$? 


**Question2:** Is there a closed set $C\subset [0,1)$ of positive measure such that $\mu(\mathbb{Q}+C)<1$, where $\mathbb{Q}$ is the set of rational numbers? 



Countable shifts of closed positive sets
The answer to question 2 (and therefore question 1 as well) is no. This follows from the [Lebesgue Density Theorem](https://en.wikipedia.org/wiki/Lebesgue's_density_theorem), which says, roughly, that a measurable set that is neither null nor co-null cannot be too evenly spread out, but it must be "lumpy" in places, like a fat Cantor set. If $\mathbb Q + D$ is measurable, then for every open interval $(a,b)$,
$$\frac{1}{b-a}\mu((a,b) \cap (\mathbb Q + D))$$
must be the same, and then the Lebesgue Density Theorem implies that it must always be either $0$ or $1$. 


