# support of integrable function is $\sigma $-finite

Theroem - Let $(X,\mathcal{B},\mu )$ be a measure space^{} and $f:X\to \u2102$ a measurable function^{}. If $f$ is integrable, then the support of $f$ is $\sigma $-finite (http://planetmath.org/SigmaFinite).

It follows easily from this result that any function in an ${L}^{p}$-space (http://planetmath.org/LpSpace), with $$, must have $\sigma $-finite support.

*:* Let ${A}_{0}:=[1,\mathrm{\infty}[$, and for each $n\in \mathbb{N}$ let ${A}_{n}:=[\frac{1}{n+1},\frac{1}{n}[$. Since $f$ is integrable, we must necessarily have $$ for each $n\in \mathbb{N}\cup \{0\}$, because

$$ |

Since $f$ and $|f|$ have the same support, and the the support of the latter is $\mathrm{supp}|f|={\displaystyle \bigcup _{n=0}^{\mathrm{\infty}}}{|f|}^{-1}({A}_{n})$, it follows that the support of $f$ is $\sigma $-finite. $\mathrm{\square}$

Title | support of integrable function is $\sigma $-finite |
---|---|

Canonical name | SupportOfIntegrableFunctionIssigmafinite |

Date of creation | 2013-03-22 18:38:47 |

Last modified on | 2013-03-22 18:38:47 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 26A42 |

Classification | msc 28A25 |

Related topic | SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable |

Defines | ${L}^{p}$ functions have $\sigma $-finite support |