I have come across several works that use "on the event" to relate a proposition involving random variables to an event, but I am confused about the precise meaning of it. For example, I am struggling to interpret (C1) in Theorem 1 from https://www2.math.upenn.edu/~pemantle/papers/jeeff.pdf (copied below). In this case, $\mathbf{E}(X_{n+1}-X_n \mid \mathcal{F}) \leq -a$ and $\{X_n > J \}$ are the proposition and event of interest respectively.
Let $X_n$ be random variables and suppose that there exist constants $a > 0, J, V < \infty$, and $p > 2$, such that $X_0 \leq J$, and for all $n$, $$ \mathbf{E}(X_{n+1} - X_n \mid \mathcal{F}_n ) \leq -a \text{ on the event } \{ X_n > J \} \quad \text{(C1)}$$ and $$ \mathbf{E}(|X_{n+1}-X_n|^p \mid X_0,\ldots,X_n) \leq V \quad \text{(C2)}$$ Then for any $r \in (0,p-1)$ there is a $c=c(p,a,V,J,r)>0$ such that $\mathbf{E}(X_n^+)^r < c$ for all $n$.
Note that $\{\mathcal{F}_n\}$ is the filtration to which $\{X_n\}$ is adapted to.
I have two ideas about the exact meaning of (C1). For both interpretations, let $(\Omega,\mathcal{F},P)$ be the probability space, and define $A_n:= \{ \omega \in \Omega : \mathbf{E}(X_{n+1} - X_n \mid \mathcal{F_n})(\omega) \leq -a \}\in \mathcal{F}$, and $B_n := \{ \omega \in \Omega : X_n(\omega) > J \} \in \mathcal{F}$.
The first idea is to interpret (C1) from an almost surely, conditional probability perspective as $$ P( A_n | B_n ) = 1.$$ However, this definition runs into problems when $B_n = \emptyset$.
The second idea is to interpret (C1) from the perspective of subsets of the sample space as $$ B_n \subseteq A_n , $$ which should be the same as $\forall \omega \in B_n: \mathbf{E}(X_{n+1} - X_n \mid \mathcal{F_n})(\omega) \leq -a$.
Are one of these interpretations the correct way to view (C1), or have I completely missed the mark here and the meaning is something different? More generally, what is the correct way to interpret "on the event"? I have seen similar statements show up in other works, relating a proposition involving random variables to an event. Sometimes they used "conditioned on the event" instead, or even "on the set".
I think you might be overthinking what "on the event" means. You can interpret it as follows: if $\omega \in \{X_n > J\}$, then $\mathbb{E} (X_{n+1} - X_n \, | \, \mathcal{F}_n)(\omega) \leq -a$.