Let $S,T\in\mathbb R$ such that $0\leq S If I am not mistaken, it is equivalent to consider
$$
m(t,s) = \int_\Omega (X(t,\omega)-X(s,\omega))^2\mathsf P(d\omega)
$$
such that $\lim\limits_{t\to s}m(t,s) = 0$ for all $s\in \mathbb R$, and to ask if
$$
\sup\limits_j\sup\limits_{t\in [t_j^n,t_{j+1}^n)}m(t,t_j)
$$
can be made small by taking $\mu_n$ small enough.
Continuity of the function under the integral
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real-analysis
stochastic-processes
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0@Elias: in first thing the integral $\|X-X^n\|_{L^2([S,T]\times\Omega)}$ depends on the partition and so we want it to go to zero with partition becoming smaller and smaller. What is unclear with the second part? – 2012-02-25