# Lebesgue measure

Let $S\subseteq \mathbb{R}$, and let ${S}^{\prime}$ be the complement^{} of $S$ with respect to $\mathbb{R}$. The set $S$ is said to be *Lebesgue measurable* if, for any $A\subseteq \mathbb{R}$,

$${m}^{*}(A)={m}^{*}(A\cap S)+{m}^{*}(A\cap {S}^{\prime})$$ |

where ${m}^{*}(S)$ is the Lebesgue outer measure of $S$. If $S$ is Lebesgue measurable, then we define the *Lebesgue measure* of $S$ to be $m(S)={m}^{*}(S)$. The Lebesgue measurable sets include open sets, closed sets^{} as well all the sets obtained from them by taking countable^{} unions and intersections^{}. However, with aid of the axiom of choice^{} it is possible to construct non-measurable sets.

The Lebesgue measure on ${\mathbb{R}}^{n}$ is the completion^{} (http://planetmath.org/CompletionOfAMeasureSpace) of the $n$-fold product measure^{} of the Lebesgue measure on $\mathbb{R}$.

The Lebesgue measure is a formalization of the intuitive notion of length of a set in $\mathbb{R}$, an area of a set in ${\mathbb{R}}^{2}$ and volume in ${\mathbb{R}}^{3}$, etc. It obeys many properties one would expect from these intuitive notions, such as invariance under translation^{} and rotation.

The Lebesgue measure was introduced by Henri Lebesgue in the first decade of the twentieth century. It became the prototypical example of what later became known simply as measure^{}, a concept which unified such diverse objects as area, probability, and function.

Title | Lebesgue measure |

Canonical name | LebesgueMeasure |

Date of creation | 2013-03-22 11:48:24 |

Last modified on | 2013-03-22 11:48:24 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 16 |

Author | bbukh (348) |

Entry type | Definition |

Classification | msc 28A12 |

Classification | msc 26A42 |

Classification | msc 03B52 |

Classification | msc 03B50 |

Related topic | Measure |

Related topic | LebesgueOuterMeasure |

Related topic | Integral2 |

Related topic | MikowskiInequality |

Related topic | VitalisTheorem |

Related topic | BorelSigmaAlgebra |

Related topic | HausdorffMeasure |

Defines | Lebesgue measurable |