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additivity is too strong; for example, it would imply that if the measure of a set. consisting of a single point is zero, then the measure of every subset of Rn would. be zero. It is not possible to deï¬ne the Lebesgue measure of all subsets of Rn in a. geometrically reasonable way. Hausdorï¬ (1914) showed that for any dimension. n â¥ 1, there is no countably additive measure deï¬ned on all subsets of Rn that is. invariant under isometries (translations and rotations) and assigns measure one to.
a straight path as shown in âgure 3.4. With functions of one variable, one way to show a limit existed, was to show that the limit from both directions existed and were equal ( lim x!a f(x) = lim x!a+ f(x)). Equivalently, when the limits from the two directions were not equal, we concluded that the limit did not exist. For functions of several variables, we would have to show that the limit along every possible path exist and are the same. The problem is that there are inânitely many such paths. To show a limit does not exist, it is still enough to ând two paths along which the limits are not equal. In view of the number of possible paths, it is not always easy to know which paths to try. We give some