Question: Let $([0,1],\mathscr B([0,1]),\lambda)$ be the probability space where $\lambda$ is th Lebesgue measure and $\mathscr B([0,1])$ is the Borel $\sigma$-algebra of the unit interval $[0,1]$. Let us define a new probability space $(\Omega,\mathscr F,\mathsf P)$ where
$$
 \Omega = [0,1]^\mathbb N
$$
and $\mathscr F$ is its product $\sigma$-algebra, and $\mathsf P$ is its product measure. I wonder if there is a standard name either for the measurable space $(\Omega,\mathscr F)$ - something like Hilber cube - or even for the whole probability space $(\Omega,\mathscr F,\mathsf P)$.

Answer: 
I've certainly seen this construction several times without any special name given to it. So while I can't say that there is no special term for it, I'm pretty sure you don't break any convention by not naming it a certain way.
I believe that in the case where $\Omega =\{0,1\}$ (the coin flipping case) Tao calls this construction a "Bernoulli Cube" in his ["Introduction to measure theory", p. 241.](http://terrytao.files.wordpress.com/2011/01/measure-book1.pdf)