Lebesgue measure vs. box counting dimension

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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 of a bounded, closed set is the same as its Lebesgue measure
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?
 

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mathman
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I am not familiar with box counting dimension. However, Lebesgue measure is not at all the same as dimension. You can have measure defined for n-dimensional spaces of any dimension or even on more abstract spaces where dimension isn't even defined.

For one dimensional space, the simplest version starts with intervals and define measure as the length of the interval. It is then extended to other sets (sigma field) using countable unions and intersections.
 

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