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This problem is in a set of old past qualifying tests I am using for practice.

Suppose $g$ is bounded measurable function of period $T$ with $m(x:|f(x)|>0)>0$. Let $(a_n)$ and $(b_n)$ numerical sequences and suppose that $G(x)=\sum_n|b_n g(nx+a_n)|$ converges for all points $x$ in a set of positive (Lebesgue) measure. Show that $\sum_nb_n$ converges absolutely.

I define $E=\{x\in\mathbb{R}: G(x)<\infty\}$ and $E_k=\{x\in \mathbb{R}: G(x)\leq k\}$. Then $E=\bigcup^\infty_{k=1}E_k$ which implies that there is $k'$ such $m(E_{k'})>0$. From this I obtain that \begin{align*} \int_{E_{k'}}G(x)\,dx=\sum_n|b_n|\int_{E_{k'}}|g(nx+a_n)|\,dx<k'm(E_{k'}) \end{align*}

This is as far I can go. I will appreciate any hint.

Mittens
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1 Answers1

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Since $m(E_{k'})>0$, there is a measurable subset $F\subset E_{k'}$ such that $0<m(F)<\infty$. Let $c_n=\int_F|g(nx + b_n)|\,dx$ (each $c_n$ is finite since $g$ is bounded and $F$ is chosen to have finite measure).

From the bound in the OP, $\sum_n|b_n c_n|<k\,m(F)<\infty$. The sequence $c_n$ is bounded below for all $n$ large enough since
$$c_n\xrightarrow{n\rightarrow\infty}\Big(\frac1{T}\int^T_0 |g(x)|\,dx\Big)\,m(F)>0$$ This is Fejer's Lemma, a result similar to the Riemann-Lebesgue lemma. The conclusion then follows by comparison.

Mittens
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