1
$\begingroup$

A set $S$ has measure zero if for any $\epsilon>0$, we can choose a collection of intervals $I_1=(a_1,b_1)$, $I_2=(a_2,b_2)$, $\ldots$ such that $S$ is contained in the union $\bigcup_{k=1}^\infty I_k$ of all the intervals, and such that the sum of the lengths of all the intervals $$\sum_{k=1}^\infty m(I_k)=\sum_{k=1}^\infty (b_k-a_k),$$ is less than or equal to $\epsilon$.

My question: Can we define measure zero on a metric space?

  • 1
    You can define measure zero on any measure space. The one you're looking at in particular is the Lebesgue measure on the real numbers, but there are many more examples of measure spaces, some of which are indeed metric spaces – 2017-01-12
  • 0
    Also this may be of interest to you: http://math.stackexchange.com/questions/987179/lebesgue-measure-on-metric-spaces?rq=1 – 2017-01-12
  • 0
    You may want to direct some attention to Hausdorff measures. They're a class of measures defined on suitably "small" metric spaces, and include Lebesgue measure on $\mathbb{R}^n$ as a special case. – 2017-01-12

0 Answers 0