Let's consider a set $A$ and another set $B$ where $B \subset T$ . Conside another set $C= T \backslash B$(exclude set B from T). Now, We are given a stochastic process $X(t)$ such that $P(X(t)_{t \in C} \in \omega) \leq f(u)$ . $f(u)$ tends to zero faster than $P(X(0) \in \omega) $ and $P(X(t)_{t \in B} \in \omega) \geq P(X(0) \in \omega )$. How can we prove that $$ P(X(t)_{t \in C} \in \omega) =o (P(X(t)_{t \in B} \in \omega)) $$ My answer:
since, $f(u)$ tends to zero faster than $P(X(0) \in \omega)$. so,
$P(X(0) \in \omega) \geq f(u) \geq P(X(t)_{t \in C} \in \omega) $ and $P(X(t)_{t \in B} \in \omega) \geq P(X(0) \in \omega )$. so,
$$ P(X(t)_{t \in B} \in \omega) \geq P(X(0) \in \omega) \geq f(u) \geq P(X(t)_{t \in C} \in \omega) $$ which does not seem to prove the result or does it?