I'm trying to teach myself stochastic processes 'properly' following a very 'physicist' education.
I'm reasonably happy with most of the basics, but something I am tripping up on is the construction of (pushforward) probability spaces of functions on uncountable index sets (i.e. continuous time).
So let's say I have a stochastic process $\{X_t\}_{t\in T}$ on the underlying probability space $(\Omega,\mathcal{F},P)$ (with samples $\omega$) which takes values in some state space $(S,\mathcal{S})$. My goal is to define/understand the law of the stochastic process $(\Omega_X,\mathcal{F}_X,P_X)$ and its relation to the product space $(S^T,\mathcal{S}^T)$.
Following Billingsley (Probability and Measure, Wiley), as I understand it, if one sets up the finite dimensional distributions
$$P_{1,\ldots,n}(X_{t_1}\in A_1,\ldots,X_{t_n}\in A_n)=P(\{\omega\in \Omega : X_{t_1}\in A_1,\ldots,X_{t_n}\in A_n\})$$
this implies the existence of a stochastic process via Kolmogorov's extension theorem (given consistency requirements etc.). However, the process is not fully defined unless you further define the nature of the paths in $X$. As I understand it we can do this with the finite dimensional distribution if we insist that the process is separable, which might mean restricting the space of functions to the set of continuous functions on $T$ (for example).
This tells me that I could have, for example if $(S,\mathcal{S})=(\mathbb{R},\mathcal{B})$ and then specify $\Omega_X=C[T,\mathbb{R}]\subset \mathbb{R}^T$, but when it comes to identifying the appropriate sigma algebra, $\mathcal{F}_X$, I realise I am missing something.
This is because, taking the above example, it is emphasised that whilst we must specify more precisely $\Omega_X$ to be $C[T,\mathbb{R}]$ in order to define the stochastic process, the continuous functions in $C[T,\mathbb{R}]$ are not in $\mathcal{B}^T$.
So my question(s) is (are) pragmatic:
I want to set up a stochastic process by defining its state space, the regularity of its paths and its finite dimensional distributions, I then get stuck at:
Say I wish to do the above for a measure on continuous functions such that $\Omega_x=C[T,\mathbb{R}]$, what is the sigma algebra, $\mathcal{F}_X$, which $P_X$ can assign probability to?
Say I wish to do the above for a measure on step functions (which are cadlag), here $\Omega_x$ is the space of all such step functions. What does the sigma algebra, $\mathcal{F}_X$, look like in this instance?
Fundamentally, I am confused as to why by taking a subset of the outcomes of the product space $S^T$ in order to properly define the stochastic process leads to the associated sigma algebras from that product space, $\mathcal{S}^T$, which is inadequate, and crucially do not know how to fix that.
What I didn't pick up in Billingsley, after he pointed out that these sigma algebras are inadequate (there is a section with this exact title), is what they should then be, I either didn't notice or understand when he explained this.
I am aware this is new to me so please do point out if this is incorrect to the point of being misleading and I will try to catch up.