Is the following set measurable: $A= V \in ([0,1]-\mathbb{Q})$ where $V$ is the Vitali set and $\mathbb{Q}$ is the set of rational numbers.
2025-01-12 19:23:03.1736709783
To check whether a set is measurable.
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If you defined Vitali set as a subset of $[0,1]$ then $V \cup \mathbb[[0,1]-{Q}]=[[0,1]-Q]\cup B$, where B is some countable set and as such Lebesgue measurable. As all the finite number of components of $A$ are measurable so is $A$.