Measure between the counting measure and the Lebegue measure - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:34:09Z http://mathoverflow.net/feeds/question/11641 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure Measure between the counting measure and the Lebegue measure Sune Jakobsen 2010-01-13T09:01:19Z 2010-01-13T09:24:32Z <p>There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is there a translation invariant measure m such that for some sets with Lebegue measure 0 the m-measure is infinite and for some sets with infinite counting measure, the m-measure is 0? </p> <p>I have found one example: m(A)=0 if A is countable, and m(A)=infinite otherwise. So I will require that the measure can take the value 1.</p> <p>If such a measure exist, can we find a measure between this and the counting measure? and between this and the Lebegue measure? and so on.</p> http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure/11642#11642 Answer by Thorny for Measure between the counting measure and the Lebegue measure Thorny 2010-01-13T09:22:18Z 2010-01-13T09:22:18Z <p>Hausdorff measures of dimensions between 0 and 1 are a continuous spectrum of examples.</p> http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure/11643#11643 Answer by Pete L. Clark for Measure between the counting measure and the Lebegue measure Pete L. Clark 2010-01-13T09:24:32Z 2010-01-13T09:24:32Z <p>I believe what you are looking for is the $\alpha$-dimensional <a href="http://en.wikipedia.org/wiki/Hausdorff%5Fmeasure" rel="nofollow">Hausdorff measure</a> for some $0 &lt; \alpha &lt; 1$.</p>