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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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    @MichaelGreinecker: finally, I've found the book I had in mind. In [Markov Models and Optimization, Section 23](http://www.amazon.com/Optimization-Chapman-Monographs-Statistics-Probability/dp/041231410X) Mark Davis calls $\Omega$ a *Hilbert cube* - rather than $(\Omega,\mathscr F)$ or $(\Omega,\mathscr F,\mathsf P)$. Hence, so far I can't remember either of seeing somewhere a special name for this measurable/probaility spaces. If you post your comment as an answer, I'll accept it.2012-08-30

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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.

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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.

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    The very next example concerns the construction in the OP, and is called "the continuous cube" by Tao, however I'm not sure whether if I say that even among probabilists, I'll be understood.2013-10-21