# Lebesgue differentiation theorem

Let $f$ be a locally integrable function on ${\mathbb{R}}^{n}$ with Lebesgue measure^{} $m$, i.e. $f\in {L}_{\text{loc}}^{1}({\mathbb{R}}^{n})$. *Lebesgue’s differentiation ^{} theorem* basically says that for almost every $x$, the averages

^{}

$$\frac{1}{m(Q)}{\int}_{Q}|f(y)-f(x)|\mathit{d}y$$ |

converge to $0$ when $Q$ is a cube containing $x$ and $m(Q)\to 0$.

Formally, this means that there is a set $N\subset {\mathbb{R}}^{n}$ with $\mu (N)=0$, such that for every $x\notin N$ and $\epsilon >0$, there exists $\delta >0$ such that, for each cube $Q$ with $x\in Q$ and $$, we have

$$ |

For $n=1$, this can be restated as an analogue of the fundamental theorem of calculus for Lebesgue integrals. Given a ${x}_{0}\in \mathbb{R}$,

$$\frac{d}{dx}{\int}_{{x}_{0}}^{x}f(t)\mathit{d}t=f(x)$$ |

for almost every $x$.

Title | Lebesgue differentiation theorem |
---|---|

Canonical name | LebesgueDifferentiationTheorem |

Date of creation | 2013-03-22 13:27:36 |

Last modified on | 2013-03-22 13:27:36 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 28A15 |