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Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process. Let $S^T$ be the collection of all functions from $T$ into $S$.

  1. In Wikipedia:

    the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The process $X$ induces a function $Φ_X : Ω → S^T$, where $$ \left( \Phi_{X} (\omega) \right) (t) := X_{t} (\omega). $$ The law of the process $X$ is then defined to be the pushforward measure $$ \mathcal{L}_{X} := \left( \Phi_{X} \right)_{*} ( \mathbf{P} ) = \mathbf P \circ \Phi_X^{-1} $$ on $S^T$.

    I was wondering if the $\sigma$-algebra on $S^T$ is defined as $\{ A \subseteq S^T: \Phi_X^{-1}(A) \in F\}$?

    Formally, I can understand the meaning of pushforward measure too. But I still don't feel it natural, perhaps because I don't actually grasp the intuition about the $\sigma$-algebra and the measure $\mathcal{L}_{X}$ defined on $S^T$. So hope to see some insights here.

  2. When $(S, \Sigma)=(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, I heard about a different way but cannot remember correctly:

    $\forall n \in \mathbb{N}, t_1, \dots, t_n \in T, B_i \in \mathcal{B}(\mathbb{R}), i=1,\dots,n$, define a subset of $\mathbb{R}^T$ $$ [B_1, \dots, B_n]:= \{f \in \mathbb{R}^T: f(t_i) \in B_i, i=1,\dots,n \} $$ and the $\sigma$-algebra on $S^T$ is the one generated from all $[B_1, \dots, B_n]$-like subsets (?).

    Then define the measure on $[B_1, \dots, B_n]$ to be $$ P(X_{t_1} \in B_1, \dots, X_{t_n} \in B_n) $$ and extend this in some way to be a measure on the $\sigma$-algebra of $S^T$ (?).

    If the above way (after filling out the question marks) is correct, is it equivalent to the Wikipedia way quoted in part 1, for inducing $\sigma$-algebra and measure on $S^T$ by the process $X$?

I would also like to know some references that can address my questions. Thanks and regards!

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This is part of my effort to understand two versions of Ito's isometry – Tim Dec 10 '11 at 22:46
The second construction is called "finite-dimensional distributions". Wikipedia is perhaps not the best place to learn about this.. I would recommend reading the first two chapters of this lecture notes: – Julian Wergieluk Dec 10 '11 at 23:15
Thanks, @wergrew! – Tim Dec 10 '11 at 23:29

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