# finite sets

• Nov 9th 2009, 05:53 PM
friday616
finite sets
Prove that every finite set has measure zero.
• Nov 9th 2009, 06:07 PM
aliceinwonderland
Quote:

Originally Posted by friday616
Prove that every finite set has measure zero.

Let X be a finite set $X=\{x_1, x_2, ..., x_n\}$.

Let $\epsilon >0$ and choose a cover for X,
$\{(x_i - \frac{\epsilon}{3n}, x_i + \frac{\epsilon}{3n})\}_{i=1}^n$

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

Thus X is of measure zero.