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

Theroem - Let $(X,\mathcal{B},\mu)$ be a measure space  and $f:X\to\mathbb{C}$ 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 $1\leq p<\infty$, must have $\sigma$-finite support.

: Let $A_{0}:=[1,\infty[$, 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 $\mu\big{(}|f|^{-1}(A_{n})\big{)}<\infty$ for each $n\in\mathbb{N}\cup\{0\}$, because

 $\displaystyle\mu\big{(}|f|^{-1}(A_{n})\big{)}\cdot\frac{1}{n+1}\leq\int_{|f|^{% -1}(A_{n})}|f|\;d\mu\leq\int_{X}|f|\;d\mu<\infty.$

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

Title support of integrable function is $\sigma$-finite SupportOfIntegrableFunctionIssigmafinite 2013-03-22 18:38:47 2013-03-22 18:38:47 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 26A42 msc 28A25 SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable $L^{p}$ functions have $\sigma$-finite support