The precise statement of the lemma is the following:
Let $\mu$ be a finite positive measure and $f \in \mathcal{L}^1(S,\Sigma, \mu)$, possibly complex valued. Define \begin{equation} a_E = \frac{1}{\mu(E)}\int_Efd\mu \end{equation} and $A = \{ a_E \mid E\in\Sigma, \mu(E) > 0\}$. Then, $\mu(\{f\notin \bar{A}\})=0$.
I have the proof in my lecture notes but I am confused about some parts:
We assume that $\mathbb{C} \setminus \bar{A}$ is not empty, so that $\mathbb{C}\setminus \bar{A}$ contains some closed ball $B$ with radius $r>0$ and center $c$. Thus $|c-a| > r$ for all $a\in \bar{A}$. It suffices to show that $E = f^{-1}(B)$ has measure zero as $\mathbb{C} \setminus \bar{A}$ is at most a countable union of such balls.
So far so good, here is where I get stuck. Suppose that $\mu(E) > 0$, then \begin{equation} |a_E - c| \leq \frac{1}{\mu(E)}\int_E |f-c|d\mu \leq r. \end{equation} This is a contradiction but honestly, I do not understand how either of the above inequalities arise. Can someone please explain? Thanks!