I know that if you take the measure of the null set, the measure is 0.

But say you take a set where the interior of the set is not the empty set. Then is the outer measure of the set positive, and is there ever a case where the measure is negative? Furthermore, why does it follow that the outer measure is 0 if the set is countable? That seems counter-intuitive to me.

  • $\begingroup$ Write down the definition of outer measure. You will find that you start with a non-negative set function. The result, the outer meausre, is again a non-negative set function. So yes, it is always non-negative. $\endgroup$ – GEdgar May 6 '15 at 0:57
  • $\begingroup$ From your second question, I guess you mean Lebesgue outer measure. Again, use the definition of outer measure to see that a single point has measure zero. Then there is a theorem about countable subadditivity. $\endgroup$ – GEdgar May 6 '15 at 0:58

Hint: For the countable set $\{x_n\}$: surround the point $x_n$ with an interval of length $\epsilon/2^n$. The sum of the lengths of the intervals is $$\epsilon + \epsilon/2 + \epsilon/2^2 + \cdots = 2 \epsilon$$


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