One learns in a probability course that a (real) random variable is a measurable mapping of some probability space $(\Omega,\mathcal{A},\mathbf{P})$ into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. But as soon as one gets into topics that are a little advanced, the space $(\Omega,\mathcal{A},\mathbf{P})$ is not mentioned unless it is absolutely necessary. After a long time of frustration, I have become quite comfortable with this language. But some things still trouble me. The following kind of reasoning comes up in the book I'm reading:
The author says that $(X_i)_{i\in I}$ is a family of random variables and specifies the distribution of each random variable. Then he phrases some (random) proposition $A((X_i)_{i_\in I})$ (this is a little imprecise, I hope you get the meaning) and talks about $\mathbf{P}[A((X_i)_{i_\in I}) \text{ holds}]$.
My question: Let (\Omega',\mathcal{A}',\mathbf{P}') be another probability space and $(Y_i)_{i\in I}$ random variables such that, for each $i\in I$, the distribution of $Y_i$ is the same as the distribution of $X_i$. Is it then obvious that \mathbf{P}[(A(X_i)_{i_\in I}) \text{ holds}]=\mathbf{P}'[(A(Y_i)_{i_\in I}) \text{ holds}]?
Now my guess is that this is true, but needs a proof, which is not completely trivial in case $I$ is infinite, at least not for a beginner. However, in the book this problem isn't discussed at all. So did I miss something?
Edit:
I'm not sure whether the question was correctly understood, so I'll rephrase it a little.
Let $(\Omega,\mathcal{A},\mathbf{P})$ and (\Omega',\mathcal{A}',\mathbf{P}') be two probability spaces, $I$ a set, and $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ families of random variables on $(\Omega,\mathcal{A},\mathbf{P})$ and (\Omega',\mathcal{A}',\mathbf{P}') respectively such that, for each $i\in I$, the distribution of $X_i$ is equal to the distribution of $Y_i$. Let $J$ be a countable subset of $I$ and $B_j$ a Borel set for each $j\in J$. The question is:
Is it obvious that \mathbf{P}\left[\bigcup_{j\in J}\{X_j\in B_j\}\right]=\mathbf{P}'\left[\bigcup_{j\in J}\{Y_j\in B_j\}\right]?
The sets $B_j$ and the union over $J$ are just an example. What I mean, but cannot formalize: Let $A\in\mathcal{A}$ and A'\in\mathcal{A}' such that there is an expression for $A$ in terms of the $X_i$, and A' is given by the same expression replacing $X_i$ by $Y_i$ for each $i$. Is it obvious that \mathbf{P}[A]=\mathbf{P}'[A']?