is a union on a measure zero set of sets of measure zero has to be a zero?

Question

If we have a measure space X , a measure zero set B and for every x in B we have a measure zero set A_x. If the union of A_x is measurable set, is it has to be of measure zero?

Answer 1

No. The Cantor ternary set $C$ is a set of measure zero, yet has cardinality $2^{\aleph_0} = | \mathbb{R} |$. Taking any bijection $f : C \to \mathbb{R}$ it follows that $\{ f(x) \}$ has measure zero for all $x \in C$, yet $\bigcup_{x \in C} \{ f(x) \} = \mathbb{R}$ obviously does not have measure zero.