We need to prove that $g\in L^p \; \forall\;\;p \; \text{s.t.} \; 0
Suppose that $ (Y, M, \eta)$ is a finite measure space, $g\in L^{+}(Y,M)$ and $\Lambda_{g}(v)\leq (v^{-1}) \quad \forall v>0$, where $\Lambda_{g} : (0, \infty) \rightarrow [0,\infty]$ by: $$\Lambda_{g}(v) =\eta(\{x:g(x)>v\}) \quad\text{and}\quad\int(g(x))^{p}d\eta(x) =p \,\int_{0}^{\infty} v^{p-1}\Lambda_{g}(v)dv$$
This is what I was trying to do but I don't know whether its correct. Suppose $p=1$, observe that the above integral equation becomes $\int_{0}^{\infty} \Lambda_{g}(v)dv$ , and then I used definition of $\Lambda_{g}$. Is is the right thing to do so far?