Archives

# 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.

Related Posts

### Comparing Topologies

The Back Pocket

### Borel-Cantelli Lemma (Pictorially)

The Back Pocket

### What do Polygons and Galois Theory Have in Common?

The Back Pocket

### Necessary vs. Sufficient?

The Back Pocket

Leave a comment!