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Measure Relation Theory Measurable Functions Sigma Finite Support Lp Space

1	Measure
1	Relation Theory
2	Measurable Functions
5	Support
5	Lp Space
6	Sigma Finite

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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 is integrable, then the support of is $\sigma$ -finite.

It follows easily from this result that any function in an -space, with $1 \leq p < \infty$ , must have $\sigma$ -finite support.

Proof: Let $A_0 := [1, \infty[$ , and for each $n \in \mathbb{N}$ let $A_n:= [\frac{1}{n+1}, \frac{1}{n}[$ . Since is integrable, we must necessarily have $\mu \big(\vert f\vert^{-1}(A_n)\big) < \infty$ for each $n \in \mathbb{N} \cup \{0\}$ , because

$\displaystyle \mu \big(\vert f\vert^{-1} (A_n) \big) \cdot \frac{1}{n+1} \leq \... ...f\vert^{-1}(A_n)} \vert f\vert \;d\mu \leq \int_X \vert f\vert \;d\mu < \infty.$

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