A question about the nonmeasurable Vitali set

Question

We know by its contstruction that the Vitali set $V$ is not a lebesgue measurable set.

But from the definition of the sigma algebra $\mathcal{M}$ of the lebesgue measurable sets,every singleton or a countable subset of the Vitali set has measure 0.

Thus $\forall a \in V$ we have that $\{a\} \in \mathcal{M}$ and because $\mathcal{M}$ is a sigma algebra then must $V$ \ $\{a\}$ $\in \mathcal{M}$.

Is this somekind of paradox or am i missing something?

Answer 1

Thus $\forall a \in V$ we have that $\{a\} \in \mathcal{M}$ and because $\mathcal{M}$ is a sigma algebra then must $V$ \ $\{a\}$ $\in \mathcal{M}$.

You would have $V$ \ $\{a\}$ $\in \mathcal{M}$ provided $V\in\mathcal M$, but this is not true.