On the associativity of the product of measure spaces (Q1416127)

The product the author is concerned with might be called the Carathéodory product of a family \((X_i,\mathcal A_i,\mu_i)\), \(i=1,\dots,n\), of positive measure spaces. Denote by \((X,\mathcal A,\mu)\) this product and fix a partition of the interval \([1,n]\) into \(p\) consecutive subintervals. Take the corresponding \(p\) products and then their product, say \((Y,\mathcal B,\nu)\). With an obvious identification, we have \(X=Y\) and, according to the main result of the paper, \(\mathcal A=\mathcal B\). However, \(\mu=\nu\) need not hold if the \(\mu_i\) are not \(\sigma\) finite. Rewiewer's remark: Another product of positive measure spaces is discussed in \textit{S. K. Berberian}'s book ``Measure and integration'' (1965; Zbl 0126.08001) and an associativity property of that product is the subject of one of the starred exercises (Exercise 39.9).