Types of set sparsity
Several terms describe a set that is “small” or “negligible,” even when it has infinitely many points. “Nowhere dense” means small in a topological sense (and a meager set is a countable union of nowhere dense sets), and “measure zero” means small in a size sense. Neither implies the other. An event of probability zero is a set of measure zero for a probability measure.
Definitions
Consider a set $A \subseteq \reals$. The set $A$ has measure zero if for every $\varepsilon > 0$, it can be covered by countably many intervals with total length less than $\varepsilon$.
Consider a measure $\mu$ on a space (by default, Lebesgue measure) and a measurable set $A$. The set $A$ has measure zero if $\mu(A) = 0$.
Examples
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The set of rational numbers, $\mathbb{Q}$
$\mathbb{Q}$ has measure zero, because it is countable.
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The Cantor set
The Cantor set is nowhere dense and has measure zero, even though it is uncountable.
Non-examples
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A “fat” Cantor set
A fat Cantor set does not have measure zero, because the intervals removed in its construction are small enough that it keeps a positive length.
Properties
- Every countable set of real numbers has measure zero, and so do some uncountable sets, such as the Cantor set.
- A countable union of sets of measure zero has measure zero.
Consider a set $A$ in a topological space, with its closure $\overline{A}$. The set $A$ is nowhere dense if the interior of its closure is empty:
\[\operatorname{int}(\overline{A}) = \emptyset.\]Examples
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The Cantor set
The Cantor set is nowhere dense, because the interior of its closure is empty. The Cantor set is closed, so its closure is the Cantor set itself. After $n$ steps of the construction, what remains is a union of intervals of length $3^{-n}$, so the Cantor set contains no interval of positive length, and therefore no open set: its interior is empty.
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A “fat” Cantor set
A fat Cantor set is nowhere dense, because it contains no interval.
Non-examples
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The set of rational numbers, $\mathbb{Q}$
$\mathbb{Q}$ is not nowhere dense, because it is dense in $\reals$.
Missing definition: meager_set
Consider an event in a random experiment. The event has probability zero if its probability is $0$.
Consider a probability space with probability measure $P$, and an event $E$. The event $E$ has probability zero if $P(E) = 0$.
Examples
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The probability that a uniform random number on $[0, 1]$ is rational
The event that a uniform random number on $[0, 1]$ is rational has probability zero, because $\mathbb{Q} \cap [0, 1]$ has measure zero. The event is not impossible, however, because every rational number in $[0, 1]$ is a possible outcome.
Properties
- An event with probability zero can still happen. If $X$ is uniformly distributed on $[0, 1]$, then $P(X = 1/2) = 0$, but $X$ must take some value, and every value has probability zero.
Examples
| Example | measure zero | nowhere dense | meager_set | probability zero | Why |
|---|---|---|---|---|---|
| The Cantor set | ✓ | ✓ | ✓ | – | The Cantor set is nowhere dense and has measure zero, even though it is uncountable. |
| The set of rational numbers, $\mathbb{Q}$ | ✓ | ✗ | ✓ | – | |
| A “fat” Cantor set | ✗ | ✓ | ✓ | – | |
| The probability that a uniform random number on $[0, 1]$ is rational | – | – | – | ✓ |
Notes
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A countable union of nowhere dense sets is called meager (or of first category). The Baire category theorem says $\reals$ is not meager.
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“Almost every” means “except on a set of measure zero,” and “almost never” means “with probability zero.”