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This is part of the proof in the construction of arbitrary product space of probability spaces given in Bauer's Probability Theory. Let $J_k$ be a nonempty finite index subset of an arbitrary index set $I$, and $J_k \subset J_{k+1}$ for all $k\in \mathbb{N}$. Let $p_{J_k}^{J_{k+1}}$ be the proje...