Re: Sets of measure 0 question...
On 12-05-2005 13:20, James wrote:
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?
If A has measure 0, you can find, for each natural n, a countable
collection A_n of open intervals such that A is a subset of the union of
the elements of A_n and that the sum of the lengths of the elements of
A_n is smaller than 1/2^n. You can always do this in such a way that
no element of an A_n belongs to some A_k with k < n. Now, take the union
of all A_n's. The sum of their lengths is smaller than 1. On the other
hand, if x belongs to A, then each A_n has an interval that contains x
and therefore the union of the A_n's has infinitely many elements that
contain x.
Best regards,
Jose Carlos Santos
.
