i am studying Random Process and i am reading the book of Durrett.When i was studying the Brownian Motion in Chapter 8,i am interested in the construction of Brownian Motion in this book. In my opinion,the construction in this book include 3 steps:
Step 1:Construct a random process $B_t$ on $Q_2$ by the Guassian tranformation function.
Step 2:Using Borel-Cantelli Lemma to prove the uniform continuity of $B_t$ on $Q_2\bigcap[0,T]$
Step 3:Extend the measure to the continuous function space.
Clearly,the author choose a countable dense subset of $B_t$ and use Borel-Cantelli Lemma to prove the uniform continuity of $B_t$ on $Q_2$.I wonder if we don't restrict our choice of $Q_2$ ,can we prove the uniform continuity in a weaker hypothesis? The clear statement of my question is as follows:
Suppose $B_t,t\geq 0$ is a real-value random process,satisfy that:
(i)If $t_0<t_1<...<t_n$,then $B(t_0),B(t_1)-B(t_0),...,B(t_n)-B(t_{n-1})$ is independent.
(ii)$\forall t\geq0,\forall 0<a<\frac{1}{2},\lim\limits_{s\rightarrow t}\frac{B_s-B_t}{(s-t)^a}\overset{P}{\rightarrow}0,\forall 0<a<\frac{1}{2}$
Prove or give a counterexample to show that there is a countable dense subset $E\subset R_+$,such that:$$\forall t\in E,\lim\limits_{s\in E,s\rightarrow t}B_t-B_s=0,a.e.$$