Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which defines two random variables
$$ X \colon \Omega \to \mathbb{R}\\ Y \colon \Omega \to \mathbb{R} $$
Now suppose that $X$ and $Y$ are independent meaning that $\mathbb{P}(\omega \in \Omega \colon X \in A, Y \in B) = \mathbb{P}(\omega \in \Omega \colon X \in A)\, \mathbb{P}(\omega \in \Omega \colon Y \in B)$ for every $A,B \in \mathcal{B}$.
Now let us duplicate the space $(\Omega, \mathcal{F}, \mathbb{P})$ yielding $(\Omega', \mathcal{F}', \mathbb{P}')$ such that $\Omega = \Omega'$, $\mathcal{F}=\mathcal{F}'$, and $\mathbb{P} = \mathbb{P}'$. We consider the product space $(\Omega \times \Omega', \mathcal{F}\otimes \mathcal{F}', \mathbb{P}\,\mathbb{P}')$.
We then extend those random variables to that product space by
$$ \overline{X}(\omega,\omega')\colon \Omega \times \Omega' \to \mathbb{R} = X(\omega), \\ \overline{Y}(\omega,\omega')\colon \Omega \times \Omega' \to \mathbb{R} = Y(\omega') $$
We can verify that $\overline{X}$ and $\overline{X}$ are random variables on that product space, and they are independent.
Am I on a correct path to think about the independence of $X$ and $Y$ in this product measure sense? In short, suppose we have two indepdendent random variables on a probaiblity space, can I think them as the sliced random variables on their product measure space?