An integrable perturbation cannot repair nonintegrability ($u\in L^1⟹(u+v\in L^1⟺v\in L^1)$)

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 $|u+v|\le |u|+|v|$; the other follows from $|v|\le |u+v|+|u|$. 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.