Consider a stochastic process $Y_k:=x+k-(X_1+\dots +X_k)$ for $k=1,2,...$, where $x \in \mathbb{N} $ and $X_1$ are i.i.d. random variables which are integer valued and non negative. The process starts at $Y_0=x$. $P_x$ is the measure under the condition that $Y_0:=x$
Let $(\Omega, \mathbb{F},\mathcal{F}, P_x)$ a filtered probability space where $\mathbb{F}:= \{ \mathcal{F_k}: k \in \mathbb{N}\}$, is the natural filtration generated by $X_k$, i.e. $\mathcal{F_k}= \sigma(X_s \mid s \leq k)$
Is $R_0 $ measurable, although my sigma-Algebra starts at $k=1$?
Since my process starts at $k=1$, I know $R_0$ by computing $P_x(Y_k=s)$ for any $k=1,2,...$
Is this sufficient?
It depends on what you are attempting to model.
For example, I've come across some stochastic processes where the paths have to be cadlag and the process predictable with regards to its underlying filtered sigma algebra.
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@Sarah; It sounds like your comments are along the right lines. I can't be more specific since its been about a decade since I've looked at this kind of material. I'd add though, that often the measure space is usually the Borel algebra of the space in question; that is one takes the minimal Borel algebra generated by the opens of the underlying topological space of the process in question.