$\mathbf{Problem}$: Let $(X,\mathcal{M},\mu)$ be a finite measure space and $1 \lt q \lt \infty$. Let $A_q = \{f: X \rightarrow \mathbb{R}$ such that $f$ is $\mathcal{M}$-measurable and there exists $C \gt 0$ such that for every $K \gt 0$, $\mu(\{x:|f(x)| \gt K \}) \lt CK^{-q} \}$. Show that $A_q \subset L^p(\mu)$ for all $1 \le p \lt q$.
$\mathbf{Attempt}$: Since the function's "overhead" gets smaller and smaller, the function appears to be integrable under restrictions on $p$. However, some routine approximation for $f \in A_q$ does not seem to work.
For instance, simply approximating the function with bounded-above version of it does not work because we end up with $\mu(\{x:|f(x)| \le K \}) \gt \mu(X) - CK^{-q}$, which doesn't seem to lead anywhere.
I also tried approximating $f$ with bounded versions of it from bottom and from above but without success, i.e. consider $E_K = \{x:|f(x)| \lt 1/K \}$, $F_K = \{x:|f(x)| \le K \}$, and $G_K = \{x: 1/K \lt |f(x)| \le K \}$. Then $\mu(E_K) \gt \mu(X) - CK^{q}$ and $\mu(F_K) > \mu(X) - CK^{-q}$, and note that $G_K = F_K - E_K$. But I can't seem to establish an inequality on the $(\int_{G_K}f^p d\mu)^{1/p}$ showing that it's finite. Am I on the right track, or is there a cleaner approach to this problem? Any help is appreciated!
Hint: Given $p \in [1,q)$, we can write, for each $N \in \mathbb{N}$, $$\begin{aligned}\int_{\{x \, \mid \, |f(x)| \leq 2^{N}\}} |f(x)|^{p} \, \mu(dx) &\leq \mu(X) + \sum_{j = 0}^{N - 1} 2^{jp} \mu\{ x \, \mid \, 2^{j} \leq |f(x)| < 2^{j + 1}\} \\&\leq \mu(X) + C \sum_{j = 0}^{N - 1} 2^{jp} 2^{-(j + 1)q}.\end{aligned}$$ Observe that $\sum_{j = 1}^{\infty} 2^{(p - q)j} < \infty$. Take $N \to \infty$ to conclude.