# Two Ways to be Small

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In real analysis, there are two ways a *measurable set* $E$ can be small. Either

- the measure of $E$ is 0, OR
- $E$ is
*nowhere dense.*

*nowhere dense*means that $E$

*exists*, but there's not much to it. Much like a spider web, or an atom which is mostly empty space. (We've discussed nowhere density before.) So here's a question: Can a set be small in one sense but not the other? How about this:

*Is it possible for a set to be nowhere dense and yet have POSITIVE measure?*

*Fat Cantor Set*is a prime example. It is nowhere dense (for reasons we have mentioned before), and

*yet*it has positive measure.