A measurable function is said to be in $L^p$ space $\mathscr{L}^p(\mathscr{X}, \mathscr{A}, \mu, \mathbb{R})$ if $\int \lvert f \rvert^p d\mu < \infty$. And a measurable function has power decaying rate of $q \geq 1$ if $\limsup_{t \to \infty} t^q \mu(\{x: |f| >t\}) < \infty$, I need to find a best function $M$ such that whenever $f$ has power decaying rate of $q$, then it still $p$-integrable if $p < M(q)$. How to find this best function?
What I have tried, I have tried to rewrite $\int \lvert f \rvert^p d\mu = \int_0^{\infty} p t^{p-1}\mu(\{x: \lvert f(x) \rvert > t\})dt $(We can get this by using product measure.) So I at least get some similar expressions. But I don't know what to do next.
A best function means if I replace the function $M$ with another function $M' < M$, then this statement won't holds.
Possibly related concept: weak $L^p$ space.
Question solved.