I am reading a paper and struggling with the concept of coupling. The paper is https://arxiv.org/pdf/0907.1404.pdf (page 14).
Definition: If $Z_1:\Omega_1\to \mathbb{R}$ and $Z_2:\Omega_2\to \mathbb{R}$ are random variables on two probability spaces, then a coupling between $Z_1$ and $Z_2$ is a probability space $\Omega'$ together with two random variables $Z_1':\Omega'\to \mathbb{R}$ and $Z_2':\Omega'\to \mathbb{R}$, such that $Z_i'$ is distributed as $Z_i$.
Question: Suppose that $Z_1$ is such that $|Z_1|\le C$ almost surely, for some $C>0$. Can we couple in such a way that $|Z_1'|\le C$ almost surely?
The reason I ask this is that the argument appears a lot in proofs of the "almost sure invariance principle". Intuitively it seems like it should be true, but I am really struggling to write it in words to convince myself. Any help would be appreciated!
If you can find a coupling between $Z_1$ and $Z_2$, then the property you look for follows from the very definition: Since $Z_1'$ has the same distribution as $Z_1$, it will also have the same almost sure bound.
More precisely, $Z_1$ is a random variable on $(\Omega_1,\mathcal{F}_1, P_1)$ and $Z_1'$ is a random variable on some $(\Omega',\mathcal{F}', P')$ with
$$ P_1\circ Z_1^{-1}=P'\circ (Z_1')^{-1}.$$
In particular,
$$ P_1(|Z_1|\le C)=P'(|Z_1'|\le C),$$
so if either of these quantities is equal to $1$, then so is the other.