Definition · applies to sets

Undergraduate (measure theory)

The Lebesgue measure assigns a size to (measurable) subsets of $\reals^n$, extending length ($n = 1$), area ($n = 2$), and volume ($n = 3$) to complicated sets. More generally, a measure is a countably additive function from a collection of sets to $[0, \infty]$.

Examples

Worked examples

  • The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $\pi$ High school
    The two-dimensional Lebesgue measure of the closed unit disk is its area, $\pi$.
  • $[0, 1]$ : $1$ High school
    The Lebesgue measure of $[0, 1]$ is its length, $1$.
  • $\mathbb{Q} \cap [0, 1]$ : $0$ High school
    The Lebesgue measure of $\mathbb{Q} \cap [0, 1]$ is $0$, because it is countable: the $n$th rational number can be covered by an interval of length $\varepsilon / 2^n$, for a total length of at most $\varepsilon$.
  • The set of real numbers, $\reals$ : $\infty$ High school
    The Lebesgue measure of $\reals$ is $\infty$, because $\reals$ contains intervals of every length.
  • The set $\setdef{0, 1}$, as a subset of $\reals$ : $0$ High school
    The Lebesgue measure of $\setdef{0, 1}$ is $0$, because each point can be covered by an interval as short as we like.
  • The Cantor set : $0$ Undergraduate
    The Lebesgue measure of the Cantor set is $0$, because its construction removes intervals of total length $\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \dots = 1$ from $[0, 1]$.

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