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> Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum\_{n=1}^{\infty}\int\_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in \mathbb{R}$.
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The question came from [old qualifying exam](https://www.math.uci.edu/sites/math.uci.edu/files/2015-Spring-Real-qual.pdf).

Although not specified, in the context of the test measure on $\mathbb{R}$ is always Lebesgue measure.

I define $q=(1-\frac{1}{p})^{-1}$. The integral can be estimated by Hölder's inequality $$\begin{align}\int\_n^{n+n^{-\alpha}} |f(x+y)|dy&=\int\_{n+x}^{n+n^{-\alpha}+x} |f(y)|dy\\&=||f||\_1\\&\leq||f||\_p||1||\_q\\&=\left(\int\_{n+x}^{n+n^{-\alpha}+x} |f(y)|^pdy\right)^{\frac{1}{p}}(n^{-\alpha})^{\frac{1}{q}}\end{align}$$

Here $||\cdot||\_p$ means the norm on $L^p \left([n+x,n+n^{-\alpha}+x]\right)$.

Suppose we can prove the integral $\int\_{n+x}^{n+n^{-\alpha}+x} |f(y)|^pdy$ decays faster than $n^{-(1+\frac{1}{q})}$ then the $n^{th}$ term should decay faster than $\frac{1}{n}$ and therefore converge. However this is the farthest I can get, any idea how to proceed?