\begin{definition}[Definition:Lower Level Set]
Let $f: S \to \overline \R$ be an extended real-valued function.
Let $\alpha \in \R$.
The '''$\alpha$-lower level set of $f$''' is the set:
:$\ds \operatorname {lev} \limits_{\mathop \le \alpha} f := \set {x \in S: \map f x \le \alpha}$
\end{definition}