I am having a lack of understanding the Skorohod space considering cadlag processes.
A random variable $X$ is measurable mapping between two measure spaces say $(\Omega,\mathcal{F})\mapsto (\tilde{\Omega},\tilde{\mathcal{F}})$.
I am struggeling about the primary space and the state space in terms of the Skorohod space, when we are considering cadlag-processes. This may be trivial, but i hope you can help me out.
Lets say we have a cadlag-process $X=(X_t)_{t\geq 0}$. How can we describe the mapping in terms of the Skorohod space $D(T,\mathbb{R}^{k})$, which is defined by consisting all cadlag paths from $T$ to $\mathbb{R}^{k}$, where $T:=[0,\infty)$ We denote by $\mathcal{B}_T$ the corresponding $\sigma$-algebra ,
So can we relate $D([0,\infty)\mathbb{R}^{k}$ to $\Omega$ or $\tilde{\Omega}$? An what does terms of the canonical presentation $X(t,\omega)=\omega(t)$ change?
My attempt:
Generally we have a given probability space $(\Omega,\mathcal{F},P)$ and a cadlag process $X=(X_t)_{t\geq 0}$ defined on it. For every $\omega \in \Omega$ the mapping $X(\omega)$ defined by $$ X(\omega): T\rightarrow \mathbb{R}^{k}\\ t\rightarrow X_t(\omega) $$ is an element of $D(T,\mathbb{R}^{k})$. So we can see $X$ as the mapping $$ X:\Omega \mapsto D(T,\mathbb{R}^{k}) $$ This mapping is $(\mathcal{F},\mathcal{B}_T)$ measurable. We denote $Q:=P^{X}$ as the by $X$ induced measure on $(D(T,\mathbb{R}^{k}),\mathcal{B}_T)$.
So for every $\mathbb{R}^{k}$-valued cadlag-process $X$, we can refer the Skorohod space, as the statespace of $X$.
However $X$ can be seen as the equivalent process of $\pi:=(\pi_t)_{t\geq 0}$ on $(D(T,\mathbb{R}^{k}),\mathcal{B}_T)$; the coordinate process $\pi_t(\omega)=\omega_t$ for $\omega \in D(T,\mathbb{R}^{k})$ so that $$ \pi:(D(T,\mathbb{R}^{k}),\mathcal{B}_T)\mapsto (D(T,\mathbb{R}^{k}),\mathcal{B}_T) $$ Let $Q^{\pi}$ be the induced measure of $\pi$. Then on $((D(T,\mathbb{R}^{k}),\mathcal{B}_T,Q)$ it holds $Q^{\pi}=Q=P^{X}$ as defined above. So we can always identify a cadlag process as a random variable on the Skorohod space.
So we can use both representations
To my understanding, Skorohod works with $\Omega$ being the Skorohod space. Then canonical projection $X$ is $X_t(\omega) = \omega(t)$
You see càdlàg processes as random variables on the skorohod space. You just care about law.
That is very useful to prove existence in law of processes defined as limit of processes.