[Math] Continuous path of stochastic processes

Question

In my probability book a stochastic process is defined as a measurable map $X: \Omega \rightarrow S^T,$ where $S^T$ is equipped with the sigma algebra of cylinder events. Our professor mentioned that the canonical image space of $X$ with its sigma algebra may be a bad choice in many situations, as plenty of events are not measurable w.r.t. this sigma-algebra.

Now, a stochastic process is continuous, if
$$P(\{\omega; t \mapsto X(\omega,t) \text{ is continuous}\})=1.$$

What I don't understand is: Why is this event measurable?

Or differently: If you read such a definition of a stochastic process, what kind of definition of stochastic processes do you have in mind such that all of this makes sense?

Answer 1

You are right; the event

$$\{\omega; t \mapsto X_t(\omega) \, \text{is continuous}\}$$

is, in general, not measurable - this is because the set of continuous functions is not contained in the sigma algebra of cylinder sets, see this question.

There are at least two ways to evade this measurability issue:

1. We say that $X$ has almost surely continuous sample paths if there exists a measurable $P$-null set $N$ such that $$\{\omega; t \mapsto X_t(\omega) \, \text{is not continuous}\} \subseteq N.$$
2. Instead of considering $X$ as a mapping $X: \Omega \to S^T$ we assume that $X: \Omega \to C^T$ where $C^T$ denotes the set of continuous functions $f: T \to S$. Moreover, we endow $C^T$ with the trace $\sigma$-algebra of the cylinder sets.