Sets of measure 0 question...

Hi, I'm having trouble with the following :

Prove that a set A in R has measure 0 if and only if there
exists a countable collection of open intervals the sum of whose lengths
is finite such that each point of A belongs to infinitely many of
the intervals.
(I can do the forward implication pretty easily if A is a countable set, but
not in general)

Any thoughts?

James


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