When summation of two sequences is finite, is one finite?

Question

$|\cdot|$ is Lebesgue measure.

Let $w(\alpha) := |\{x:f(x)>\alpha\}|$

Let $f$ be a nonnegative function.

Then, the proof uses that $\displaystyle\sum_{k=-\infty}^{\infty} 2^{kp}w(2^k)\lt\infty \Longrightarrow w(\alpha)<\infty \text{ in } (0, \infty)$.

Can someone explain why $w(a)$ is clearly finite in $(0, \infty)$?

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For your information, the proof is about that if $f\ge0$, then $\displaystyle f\in L^p \Leftrightarrow \sum_{k=-\infty}^{\infty} 2^{kp} w(2^k) \lt \infty$

Answer 1

The convergence of the series implies that $w(2^k)$ is finite for each integer $k$. Now, observe that if $t\gt 0$, then $2^k\leqslant t$ for some $k\in\mathbb Z$, hence $$w(t)\leqslant w(2^ k)<+\infty.$$