I am struggling on this qualifying exam question that I found. A hint is provided that says Fubini may be helpful here, but I can't see how to setup the problem to apply it. Here is the question:

Let $(X,\mathcal{M},\mu) $ be a finite measure space. Let $f:X\rightarrow \mathbb{R} $ be measurable. Suppose that there is a $R>0$ such that for all $r>R$: $$ \mu(\{x\in X:|f(x)|>r\}) = \frac{1}{r^2}$$

Find all values of $p$, with $1\leq p \leq \infty $, for which $f\in L^p(X,\mu)$.

So far I have made the following, albeit trivial, progress:

First I recognize that since $(X,\mathcal{M},\mu) $ is a finite measure space that if $1\leq p <q \leq \infty$ then $ L^q(X,\mathcal{M},\mu) \subset L^p(X,\mathcal{M},\mu)$, so my goal is to see find the largest $p$ such that $f\in L^p(X,\mathcal{M},\mu) $.

Let $Y_r=\{x\in X :|f(x)|>r \} $, then $$\int_X |f(x)| d\mu(x)=\int_{X\setminus Y_r}|f(x)|d\mu(x) + \int_{Y_r} |f(x)| d\mu(x) $$ where the first integral on the RHS is finite since it is at most $r\cdot\mu(X)<\infty $ and perhaps I can show that the latter integral on the RHS is finite for some fixed $r\in \mathbb{R}$, but I suspect I am on the wrong path.

How would one go about solving this question using Fubini's Theorem?