Prove that every finite set has measure zero.

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- Nov 9th 2009, 05:53 PMfriday616finite sets
Prove that every finite set has measure zero.

- Nov 9th 2009, 06:07 PMaliceinwonderland
Let X be a finite set $\displaystyle X=\{x_1, x_2, ..., x_n\}$.

Let $\displaystyle \epsilon >0$ and choose a cover for X,

$\displaystyle \{(x_i - \frac{\epsilon}{3n}, x_i + \frac{\epsilon}{3n})\}_{i=1}^n$

Then $\displaystyle \sum_{i=1}^{n}[(x_i + \frac{\epsilon}{3n})- (x_i - \frac{\epsilon}{3n})] = n \frac{2 \epsilon}{3n}= \frac{2\epsilon}{3} < \epsilon$

Thus X is of measure zero.