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

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

    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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