= An integrable perturbation cannot repair nonintegrability
{title2=$u\in L^1\ \Longrightarrow\ (u+v\in L^1\iff v\in L^1)$}
If $u$ is <Lebesgue integrable> and $v$ is measurable on the same <measure space>, then $u+v$ is integrable exactly when $v$ is integrable. One implication follows from $|u+v|\leq|u|+|v|$; the other follows from $|v|\leq|u+v|+|u|$. Thus adding a <function> with finite <integral> of its <absolute value> cannot remove a failure of <absolute integrability>. This is the <vector space> property of <L1 space> stated as a useful test for a sum of terms with different decay rates.
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