I'm given a measurable space $(\Omega, \mathcal F)$ and two probability measures $\mu,\nu$ on it with $\mu\ne\nu$. Consider the product measures $P:=⊗_{k\in \Bbb N}\mu$ and $Q:=⊗_{k\in \Bbb N}\nu$ on $(\prod_{k\in \Bbb N}\Omega,⊗_{k\in \Bbb N}\mathcal F)$. My goal is to show that $P$ and $Q$ are singular.
Since our current topic in the lecture is related to convergence of random variables I assume that possible solution might deal with the Laws of Large Numbers. However, in order to apply them I would have to construct a suitable sequence of random variables and I'm really stuck with it. What would be their state space, for example ?
Any hint would be highly appreciated.