In a 2-d plane, let $A$ be the set of all points inside a circle $C$ and $P$ be the set of all points on the perimeter of the circle $C$. Is there any generalized measure in set theory which distinguishes between two such sets ?
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The area of A is positive, while the area of P is zero. Did you mean to distinguish between the interior of the circle and the closure of that set (i. e. the union of the perimeter and the interior)?