Two Ways to be Small
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.
Intuitively, to say the measure of $E$ is $0$ means that the total "length" of the "stuff" in $E$ is zero (measure = a generalization of length). To say $E$ is 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?
The answer is YES! The Fat Cantor Set is a prime example. It is nowhere dense (for reasons we have mentioned before), and yet it has positive measure.
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