The infinite union of singletons in the Lebesgue measure in $\mathbb{R}$.

Question

I know that the Lebesgue measure of a countable infinite number of singletons has a zero measure, but what about the Lebesgue measure of an infinite number of singletons, which is not countable. Can we generalize on this particular measure space? Or can some one provide a counter example?

Answer 1

Well, any subset of $\mathbb{R}$ can be written as a union of singletons, so if it was true for any such union then every measurable set would have measure zero. However, $\mathbb{R}$ doesn't have measure zero.