Re: Sets of measure 0 question...

From: "Jos Carlos Santos" <jcsantos@xxxxxxxx>
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.

> 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.

Ok, the first part of ==>.

> 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.

This is the second part of ==>?
This I don't see. What if A = N, the integers?

In fact, if sum_j |A_j| = k < oo, then for all j in N, |A_j| < k
and every A_j contains at most [A_j] <= k elements of N.

----

.

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