How to prove the following theorem by distribution function and series

Question

Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e., $$ \mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0. $$ Let $\eta>1$ and $M>1$ be constants. Then, for $0<p<\infty$, $$ g\in L^{p}(\Omega)\Leftrightarrow \sum_{k\geq1}M^{pk}\mu_{g}(\eta M^{k})=S<\infty $$ and $$ C^{-1}\leq \left\|g\right\|_{L^{p}(\Omega)}^{p}\leq\left(\left|\Omega\right|+S\right), $$ where $C>0$ is a constant depending only on $\eta, M$ and $p$.

Any help will be appreciated.