Full Measure
Undergraduate (subsets of $\reals$)
Consider a set $A \subseteq \reals$. The set $A$ has full measure if the complement $\reals \setminus A$ has measure zero.
Graduate (measure theory)
Consider a measure $\mu$ on a space $X$ and a measurable set $A \subseteq X$. The set $A$ has full measure if the complement of $A$ has measure zero:
\[\mu(X \setminus A) = 0.\]Examples
Non-examples
Things that may be mistaken for examples.
-
$\mathbb{Q}$, as a subset of $\reals$
Undergraduate
$\mathbb{Q}$ does not have full measure in $\reals$, because its complement $\reals \setminus \mathbb{Q}$ does not have measure zero.
Properties
- A set $A$ has full measure if and only if almost every point of $X$ is in $A$.
- A countable intersection of sets of full measure has full measure, because the complement of the intersection is a countable union of sets of measure zero.
- For Lebesgue measure on $\reals^n$, every set of full measure is dense, because every nonempty open set has positive measure and so is not contained in the complement.
- For a probability measure, an event has full measure if and only if its probability is $1$.
Notes
“Full measure” is about the complement, not about the measure of $A$ itself. If $\mu(X)$ is finite, then $A$ has full measure if and only if $\mu(A) = \mu(X)$. If $\mu(X)$ is infinite, then $\mu(A) = \mu(X)$ is not enough: the interval $(0, \infty)$ has infinite Lebesgue measure, like $\reals$, but its complement $(-\infty, 0]$ also has infinite measure, so $(0, \infty)$ does not have full measure in $\reals$.
Distinctions involving this term
- Types of Set Density: “Dense” and “almost everywhere” both mean “nearly everything,” in different senses. A dense set is near every point; an “almost everywhere” set misses only a set of measure zero. Neither implies the other.