Let $f: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$be a decreasing, continuous probability density function and let $m_p=\int_0^{\infty} x^p f(x) d x$ be its $p$ th moment. Show that $\left((p+1) m_p\right)^{\frac{1}{p+1}}$ is increasing in $p \in[0, \infty)$.
Also there is a hint given that :Consider a measure $\nu$ such that $\nu[x, \infty)=f(x)$ and relate $m_p$ to $\nu$. But I did not get how to relate this measure with m_p,i need a bit more hint.