There are some questions I am curious to make sure. If you could provide references, I will appreciate it very much.
1.X and Y are two random variables. Whether is the following deductions correct or not:
(1)$X\mathop = \limits^d 0 \Rightarrow X\mathop = \limits^{a.s.} 0$ ------I think it is correct. Where to find the proof?
(2)$X-Y\mathop = \limits^d 0 \Rightarrow X\mathop = \limits^{d} Y$ ------I think it is wrong
(3)$X-Y\mathop = \limits^{a.s.} 0 \Rightarrow X\mathop = \limits^{a.s.} Y$ ------I think it is correct
2.{$X_t(w)$} is a stochastic process, how to understand the variable $\mathop {\sup }\limits_{0 \le s \le T} {X_s}(w)$? Do I understand it right? Fixed each w (the sample path), there is a sup among $\{X_s(w),s\le T\}$. So each w correspond to a value, which forms a random variable.
3.If {$X_t(w)$} is an adapted process, for $s
4.what's the relation between $\mathop {\sup }\limits_{0 \le s \le T} {X_s^p}(w)$ and $[\mathop {\sup }\limits_{0 \le s \le T} {X_s}(w)]^p$, for positive p and positive process $\{X\}$.