# Cube with the product uniform measure

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 $(\Omega,\mathscr F,\mathsf P)$.

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This space looks essentially like the coin flipping measure on $\{0,1\}^\mathbb{N}$. There is an isomorphism between the space of coin flips and Lebesgue measure on $[0,1]$ and obviously, $\mathbb{N}$ can be identified with $\mathbb{N}\times\mathbb{N}$. – Michael Greinecker Aug 27 '12 at 11:57
@MichaelGreinecker: thanks! Sure, but I was just wondering about the terminology - whether there is a "canonical" name for $(\Omega,\mathscr F,\mathsf P)$ – Ilya Aug 27 '12 at 12:02
The unit hypercube? – hardmath Aug 27 '12 at 13:13
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. – Michael Greinecker Aug 28 '12 at 6:52
@Ilya If I was unclar: You are in good company if you don't use a specific term for this probabiliy space. – Michael Greinecker Aug 28 '12 at 8:55

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.