Measure zero
Undergraduate (subsets of $\reals$)
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 $\varepsilon$.
Graduate (measure theory)
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
Examples
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The set of rational numbers, $\mathbb{Q}$
High school
$\mathbb{Q}$ has measure zero, because it is countable.
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The Cantor set
Undergraduate
The Cantor set is nowhere dense and has measure zero, even though it is uncountable.
Non-examples
Things that may be mistaken for examples.
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A “fat” Cantor set
Graduate
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.
Distinctions involving this term
- 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.