Help understanding the definition of the function space $\mathcal{L}^\infty(\mu)$

Question

In René Schilling's Measures, Integrals and Martingales, the function space $\mathcal{L}^\infty(\mu)$ is defined

$$ \mathcal{L}^{\infty}(\mu):=\{u: X \rightarrow \mathbb{R}: u \in \mathcal{M}(\mathscr{A}), \exists c>0, \mu\{|u| \geqslant c\}=0\}. $$

Can someone help clarify what $\mu\{|u| \geqslant c\}=0$ means? I assume, first of all, that is should be read $\mu\left(\{|u| \geqslant c\}\right)=0$, but what exactly is the set $\{|u| \geqslant c\}$?

Answer 1

The set $\{|u| \geq c\}$ is the set $\{x \in X : |u(x)| \geq c\}$.

Thus a measurable function $u \in \mathcal{L}^\infty(\mu)$ if and only if there exists $c > 0$ such that $|u(x)| < c$ for $\mu$-almost every $x$.

The smallest such $c$ is often referred to as the essential supremum of $u$ with respect to $\mu$.

Another way to think about it is that $u$ is in $\mathcal{L}^\infty(\mu)$ if and only if there exists a function $v$ which is bounded such that $u = v$, $\mu$-a.e.