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Around the turn of the century, mathematicians such as **Emile Borel** and **Henri Lebesgue** were looking for a precise **definition** of the measure of a subset of the real n-dimensional **space** R^{n}. Four considerations bounded their search. First, the measure of any 'normal' object such as a line segment in R^{1}, a **square** in R^{2}, or a cube in R^{3} should be equal to its length, **area**, or **volume** respectively. Second, if A and B are disjoint subsets then the measure of A union B should be equal to the measure of A plus the measure of B. Third, if A and B are congruent **sets** then their measures should be equal. Fourth, it should be possibly to measure 'most' sets or at least all 'nice' ones. In 1924, **Stefan Banach** and **Alfred Tarski** dramatically demonstrated that not all subsets could be measurable if...

This section contains 952 words(approx. 4 pages at 300 words per page) |