I have a question related to the definition of supremum of a set. Consider a set $A\subseteq \mathbb{R}$. My understanding is that the supremum of $A$ is by definition a finite number. Hence, we have that if and only if $A$ is bounded above then the supremum exists. Is this correct?
If this is correct, consider the definition of Lebesgue Integral for non-negative functions here https://en.wikipedia.org/wiki/Lebesgue_integration#Integration
$\int_E fd\mu:=\sup\{\int_E sd\mu \text{ s.t. $0\leq s\leq f$, $s$ simple}\}$
and then we read "For some functions, this integral $\int_E fd\mu$ is infinite".
My second question is: the set $\sup\{\int_E sd\mu \text{ s.t. $0\leq s\leq f$, $s$ simple}\}$ does not necessarily have an upper bound since $\int_E sd\mu$ could be equal to $+\infty$ or $-\infty$. Hence, how can the supremum exist? I'm really confused on this point; any hint would be really appreciated.