Short version: What is the difference between the Lebesgue measure and the box counting dimension of a set? Long version: I was reading up on the definition of the Lebesgue measure, and the description of how to take the Lebesgue measure of a set (which I understood basically as "cover the set with ranges of a certain size, then count the number of ranges in the limit as the size of the ranges goes to zero) sounded exactly like the procedure for taking the set's box counting dimension. Meanwhile I found this cryptic sentence on Mathworld: The Minkowski measure is of course the same as the box counting dimension; I'm assuming the Minkowski dimension and the Minkowski measure are the same thing (which I guess leads to another question I should be asking-- what if anything is the difference between measure and dimension?). So is it correct that the Lebesgue measure and the box counting dimension are in fact the same thing for a bounded, closed set? And if so, in what way does this fail to be the case for unbounded or open sets?