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 , but what exactly goes on on the right hand side of the equation? It looks like a multiplication of the two functions: and but the "types" don't match, since , whereas

Thanks