# Filtration exercise

I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus".

Denote by $$\mathcal{F}^X_{t_0}$$ the natural filtration corresponding to a process $$X:[0,\infty)\times \Omega \to \mathbb{R}^k$$. Suppose that every sample path of $$X$$ is RCLL (Right Continuous on $$[0,\infty)$$ and with finite Left-hand Limits on $$(0,\infty)$$)

Introduce an event $$A\subset\Omega$$ that $$X$$ is continuous on $$[0,t_0)$$ (we have an underlying probability space $$(\Omega, \mathcal{F}, P)$$). The task is to show that $$A\in \mathcal{F}^X_{t_0}.$$

I see that $$A=\left\{\omega\in\Omega\,|\, \forall t\in [0,t_0)\,:\, \lim_{s\to t^{-}} X(s,\omega) = X(t,\omega)\right\}.$$ However I have no clue how to prove that this is somehow in the minimal $$\sigma-$$algerbra $$\mathcal{F}^X_{t_0}$$ what I interpret as the minimal $$\sigma-$$algerbra spaned by the sets of the form $$\left\{\omega\in\Omega\,|\,X(t,\omega)\in B\right\}$$ where $$0\leq t \leq t_0$$ and $$B$$ is an arbitrary Borel subset of $$\mathbb{R}^k.$$

I thought to somehow interpret the limiting property $$\lim_{s\to t^{-}} X(s,\omega) = X(t,\omega)$$ with a set-theoretical approach as follows:

$$\begin{gather*} \{\omega\in\Omega\, | \, \lim_{s\to t^{-}} X(s,\omega) = X(t,\omega)\} =\\ \bigcap_{n\in\mathbb{N}} \bigcap_{m\in\mathbb{N}, m\geq m^*_n}\{\omega\in\Omega\,|\,\forall s\in(t-1/m,t)\,:\, ||X(t,\omega)-X(s,\omega)||<1/n\} \end{gather*}$$

Above the index $$m^*_n$$ comes from a usual sequential understanding of left-hand limits.

However, I am not able to make a step forward as I do not know how to deal with the continuous character of $$t,s$$ variables above (i.e. it is not easy to show measurability in $$\Omega$$ unless I can somehow make discrete unions/intersections...)

Hopefully, someone can help. Thx in advance!

Continuity at zero is included in the RCLL property. If $$X(\cdot,\omega)$$ is discontinuous at some $$t \in (0,t_0)$$, then the left and right limits at $$t$$ (which exist) must differ. Thus there must exist an integer $$n>0$$ such that $$\|\lim_{s\to t^{-}} X(s,\omega)-\lim_{s\to t^{+}} X(s,\omega)\|>1/n \,.$$ Since the limits exists, this implies that for every integer $$k \ge 1$$ there exist rational numbers $$q_k,r_k$$ such that $$(*) \; \, \; q_k1/n \; .$$
Conversely, If $$X(\cdot,\omega)$$ is continuous on $$[0,t_0)$$ then it is uniformly continuous there (since replacing the value at $$t_0$$ by the left limit yields a continuous function on a closed interval) so (*) must fail for some $$k \ge 1$$.
Therefore, $$X(\cdot,\omega)$$ is continuous on $$[0,t_0)$$ if and only if for every $$n \ge 1$$ there exists $$k \ge 1$$ such that for all pairs of rationals $$q,r \in (0,t_0)$$ with $$q, we have $$\|X(q,\omega)-X(r,\omega)| \le 1/n$$.
Writing this in set theory notation shows that $$A\in \mathcal{F}^X_{t_0}.$$