Let $X, Y$ be real-valued random variables on a probability space $(\Omega, F, P)$, and $N$ be a random variable under $(\Omega, F, P)$ which is normally distributed.(i.e. normal distribution exists in this space)
Suppose $(\Omega', F', P')$ is a coupling of $X$ and $Y$, does normal distribution exist on this space? Are there common conditions(for example on coupling or on $\Omega$) that make this true?