essential supremum of conditional distribution

Question

Let $X,Y$ be some nonnegative real random variables. I am trying to do some calculations but I am not sure about the correct usage of definitions. Is the essential supremum of $X$ given $Y$ $$\operatorname{ess sup} X\mid Y$$ given by $$\max\{a\in\mathbb R\mid E\left(\mathbb{1}_{\{X\le a\}}\mid Y\right)<1\} \text{?}$$

Answer 1

You've almost got it. $$\require{cancel} \begin{align} & \xcancel{\max\{a\in\mathbb R\mid E\left(\mathbb{1}_{\{X\le a\}}\mid Y\right)<1\}} \\[12pt] & \sup\{a\in\mathbb R\mid E\left(\mathbb{1}_{\{X\le a\}}\mid Y\right)<1\} \end{align} $$ The reason you need $\sup$ instead of $\max$ is that if $a$ is the essential supremum, then the probability that $X\le a$ is exactly $1,$ not less than $1,$ so $a$ is not a member of the set whose supremum is taken.