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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?

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Only cardinality of $B$ is relevant, not measure. –  Jonas Meyer Jan 28 '13 at 9:08

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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.

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