Maybe this is a silly question, but it's important that I understand this. I just started to read the book Level Sets and Extrema of Random Processes and Field, from Jean-Marc Azais and Mario Wschebor. In the very beginning there is this.
Denote by $M$ the range space of the process $\mathcal{X}$. From my understanding, for each $t\in T$, we have that $X(t):\Omega\to M$ is a random variable.
Fixing $u\in M$ in the range space, I can understand the notation $\{X(t) = u\}$, which is $\{\omega\in\Omega: X(t)(\omega) = u\}$ more precisely. But the set $\{t\in T: X(t) = u\}$ is strange to me. They are saying this is a set of parameters $t$ such that $X(t)=u$. But $X(t)$ is a function, not a value! If we want to write precisely what this set is, I can see two possibilities.
1) $\{t\in T: X(t) = u\} = \{t\in T: X(t) \textrm{ is a constant function such that } X(t)(\omega) = u \textrm{ for all } \omega\in\Omega \}$
2) $\{t\in T: X(t) = u\} = \{t\in T: \textrm{ exists }\omega\in\Omega \textrm{ such that } X(t)(\omega) = u \}$
Is any of this interpretations correct? If not, what is the correct interpretation?
Thank you.
Note that it is a random set, i.e., it is a function of $\omega$, or a random variable. Perhaps this notation may help you understand: $$\{t \in T : X(t) = u\}(\omega) := \{t \in T : X(t)(\omega)=u\}$$