 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[False.] Take $k(x)=1/(1+|x|)$. This is bounded and measurable on $\mathbb R$, and tends to zero as $x\to\infty$, but
$$
\int_0^Rk(x)\,dx=\log(1+R)\longrightarrow\infty.
$$
Thus decay to zero alone does not imply <Lebesgue integrability>.

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