Let $(X, \Sigma, \mu)$ be a measure space of finite measure and $f\in L^1 (\mu)$.
For every $E\in \Sigma$ we define:
$$v(E) = \int _ E f d\mu$$
Suppose that $f\in L^p (\mu)$ for some $p \in [1, + \infty)$. Show that
$$\sup _{\pi} \sum_i \dfrac{|v(E_i)|^p}{\mu (E_i) ^{p-1}} < +\infty ,$$
where the sums are countable and the elements of $\pi$ are all measurable, countable partitions of $X$ (i.e. $X= \bigcup E_i$, $E_i \in \Sigma$, $E_i \cap E_j = \emptyset$ if $i\neq j$) such that $\mu (E_i) > 0$ $ \forall i.$
I dont have any clue on how to bound all of those sums, so I would appreciate very much your help! Thanks!