# almost everywhere

Let $(X,\U0001d505,\mu )$ be a measure space^{}. A condition holds *almost everywhere* on $X$ if it holds “with probability $1$,” i.e. if it holds everywhere except for a subset of $X$ with measure $0$. For example, let $f$ and $g$ be nonnegative functions on $X$. Suppose we want a sufficient condition on functions $f(x)$ and $g(x)$ such that the relation^{}

$${\int}_{X}f\mathit{d}\mu (x)\le {\int}_{X}g\mathit{d}\mu (x)$$ | (1) |

holds. Certainly $f(x)\le g(x)$ for all $x\in X$ is a sufficient condition, but in fact it’s enough to have $f(x)\le g(x)$ almost surely on $X$. In fact, we can loosen the above non-negativity condition to only require that $f$ and $g$ are almost surely nonnegative as well.

If $X=[0,1]$, then $g$ might be less than $f$ on the Cantor set^{}, an uncountable set with measure $0$, and still satisfy the condition. We say that $f\le g$ almost everywhere (often abbreviated *a.e.*).

Note that this is the of the “almost surely” from probabilistic measure .

Title | almost everywhere |
---|---|

Canonical name | AlmostEverywhere |

Date of creation | 2013-03-22 12:20:58 |

Last modified on | 2013-03-22 12:20:58 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 60A10 |

Synonym | almost surely |

Synonym | a.s. |

Synonym | a.e. |

Synonym | almost all |