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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.
It follows easily from this result that any function in an $L^p$ -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 $f$ is integrable, we must necessarily have $\mu \big(|f|^{-1}(A_n)\big) < \infty$ 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 $\displaystyle \mathrm{supp} \, |f| = \bigcup_{n = 0}^{\infty} |f|^{-1} (A_n)$ , it follows that the support of $f$ is $\sigma$ -finite. $\square$
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