I'm currently studying real analysis and the topic of essential supremum came up. My book did not mention anything beyond a definition. I tried looking through online resources but most just give the definition and the two same basic examples listed on wiki so I'm struggling a little bit. I'm working with the definition Given $(X, \mathcal{S},\mu)$ a measure space, essential$\sup f := \inf \{a \in \mathbb{R} : |f(x)| \leq a \; \text{for almost every x}\}$, i.e. $\mu(\{x : |f(x)| > a\}) = 0$
I'm trying to work through a few examples on my own but isn't sure if I'm understanding correctly.
For $f(x) = x^2 + ix$ for $X = [0, 10]$ with $i$ being the imaginary $i$ and Lebesgue measure. Since $f(x)$ is finite everywhere on $X$, I took $a = f(10) = 100 + 10i$ and clearly $f(x) \leq a$. Then consider the set $\{x : f(x) > a\} = \emptyset$, then $\mu(\{\emptyset\}) = 0$ so essential $\sup f = a = 100 + 10i$. This one I have a feeling is wrong since the set of $x$ where the measure is $0$ is not in the given set $X$ which is a closed interval. I assumed by the definition that the set of $x$ where the measure is $0$ has to also be in the given set $X$. Or would essential $\sup f = \infty$ since there is no such $a$?