i am working through a proof that there could be no measure on $\mathbb{R}$ such that
- $\lambda([a,b]) = b - a$
- $\lambda(A) = \lambda(A + \{c\})$
First a set $A \subset [0,1]$ is constructed with $$ \forall x \in [0,1] ~ \exists ! y \in A ~:~ x - y \in \mathbb{Q} $$ (the notation $\exists !$ means there exists exactly one such element) Then the following set is considered $$ B := \bigcup_{r \in [-1:1] \cap \mathbb{Q}} (A + \{r\}) $$ and it is claimed that it has the property $$ [0,1] \subset B \subset [-1:2] $$ but i don't see why this inclusion-relations hold?