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I was referring a section from ' CALCULUS' Volume I (Apostol) where it was mentioned that a collection of sets in a plane to which area can be assigned are called measurable sets. So immeasurable sets are collection of sets to which area cannot be assigned? Is it as simple as that or more meaning can be assigned to those sets.

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In a simplistic way, one wishes to assign a notion of "measure" (volume, area, what have you) to subsets of $\mathbb{R}^n$. Of course, being creatures of habit, we'd like this notion of measure to satisfy certain properties--axioms if you will. Things like, perhaps, the measure of the disjoint union of two sets (i.e. the union two non-intersecting sets) to be the sum of their measures, etc. People then realized that it was impossible to define a notion of measure for which all the axioms we'd like to be true are true AND is defined on all subsets of $\mathbb{R}^n$. Being those creatures of habit again, we are unwilling to give up our beloved axioms, and so to create such a measure we need to concede that we won't be able to "measure" all the subsets of $\mathbb{R}^n$. These are the "immesurable" subsets. See herefor more detail