In a book I am studying it states that a condition on a function $g$ as follows:
Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| \leq s}|g(x,u)| \leq h_{s}(x)$$ for a.e. $x \in \Omega$, all $s > 0$ and some function $h_{s} \in L^{\frac{1}{1-\epsilon}}(\Omega)$, $0 < \epsilon < 1$.
Why can't this simply be written as:
Given a function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function such that $$|g(x,u)| \leq h(x)$$ for some $h \in L^{\frac{1}{1-\epsilon}}(\Omega), 0 < \epsilon < 1$?
Thanks