I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find another sequence $v_n$ (on another probability space), which has the same distribution as the original approximate solutions $u_n$, like (1) below. Moreover, $v_n$ is a convergent sequence. One can prove the new sequence $v_n$ also satisfies the approximate scheme and then take limit to get a martingale solution $v$ to the target SPDE. But how can we know the initial distribution of $v(0)$ is the same as the given initial distribution? For example, does (3) below hold true particularly for $t=0$???. Many papers on different models omit this, so it seems very easy. Maybe I am so stupid to know why. If necessary, I can give the references to focus on a specific example. I also want to know wether we can have (2).

Let $(\Omega_i, \mathcal{F^i},\mathbb{P},\{\mathcal{F^i}_t\}_{t\geq0})$ be two probability space. Let $T>0$. Let $u:\Omega_1\times[0,T]\mapsto X$ and $v:\Omega_2\times[0,T]\mapsto X$ be two $X$-valued process such that $u\in L^2\left(\Omega_1;C([0,T];X)\right)$ and $v\in L^2\left(\Omega_2;C([0,T];X)\right)$.

Assume that $$\mathbb{P_1}(u\in A)=\mathbb{P_2}(v\in A)\ \ \ \forall A\in\mathcal{B}(C([0,T];X)),\ \ \ (1)$$ where $\mathcal{B}(C([0,T];X))$ denotes the Borel sets of $C([0,T];X)$.

My questions are:

(a) For any $t\in[0,T)$, can we have $$\mathbb{P_1}(u\in A)=\mathbb{P_2}(v\in A)\ \ \ \forall A\in\mathcal{B}(C([0,t];X))\ \ ???\ \ \ (2)$$

(b) For any $t\in[0,T]$, can we have $$\mathbb{P_1}(u(t)\in A)=\mathbb{P_2}(v(t)\in A)\ \ \ \forall A\in\mathcal{B}(X)\ \ ???\ \ \ (3)$$