TITLE: Is the countable union of measure-zero sets zero?
QUESTION [5 upvotes]: Let $E=\cup_{i=1}^\infty A_i$ where the measure of $A_i$ is zero. How can I conclude that $E$ has measure zero?

REPLY [5 votes]: If your measure $\mu$ is  sub sigma additive you can surely say that 
\[ \mu(E)\leq \sum_{i=1}^\infty \mu (A_i)=0\]
If it's not the statement doesn't need to be true.