Unclear equation in a proof in Kallenberg's Foundations of Modern Probability

**Mishmish** · July 22nd 2012, 05:28 AM

Kallenberg, "Foundations of Modern Probability", 1st edition, 1997

I don't get equations (15) in the proof of Theorem 5.17 "extension by conditioning, Ionescu Tulcea" (p. 93, Ch. 5 "Conditioning and Disintegration", see attachment).

Equation 15 is meant to define $f^{n}_{k}$ , but what exactly goes on on the right hand side of the equation? It looks like a multiplication of the two functions: $\mu_{k+1}\otimes\cdots\otimes\mu_{n}$ and $1_{A_{n}}$ but the "types" don't match, since $(\mu_{k+1}\otimes\cdots\otimes\mu_{n})\in(S_{1} \times\cdots\times S_{k})\times(\mathcal{S}_{k+1}\otimes\cdots\otimes \mathcal{S}_{n})\rightarrow\mathbb{R}$ , whereas $1_{A_{n}}\in\mathfrak{F}_n\rightarrow\mathbb{R}(=( S_1\times\cdots\times S_n)\rightarrow\mathbb{R})$

Thanks

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