I am struggling with section 18 of the textbook "Probability and Measure" by Billingsley. In particular with the construction of product measures. The contruction uses two measure spaces $(X,\mathcal X,\mu)$ and $(Y,\mathcal Y,\nu)$ where $\mu$ and $\nu$ are assumed finite. In short, the argument is the following:
- since the section of $E \in \mathcal X \times \mathcal Y$ is measurable, the measure of the section, i.e. $\nu[y:(x,y) \in E]$ is a well-defined function of x.
- the class of $E \in \mathcal X \times \mathcal Y$ for which such function is measurable is a $\lambda$-system and contains the $\pi$-system of measurable rectangles, therefore it coincides with $\mathcal X \times \mathcal Y$ (because the latter is generated by measurable rectangles)
- it is stated that $\pi'(E)=\int_X \nu[y:(x,y) \in E] \mu(dx)$ is a finite measure on $\mathcal X \times \mathcal Y$. For the other section the measure is denoted by $\pi''(E)$
- For measurable rectangles $\pi'(A\times B) = \pi''(A\times B) = \mu(A)\nu(B)$, therefore the class of $E \in \mathcal X \times \mathcal Y$ for which $\pi'(E) = \pi''(E)$ contains the measurable rectangles, and since it is a $\lambda$-system it contains $\mathcal X \times \mathcal Y$.
The result is then extended to $\sigma$-finite measures. I know that the $\sigma$-finite assumption is necessary (popular example here) but I can't see where the finiteness assumption is required in the argument above: could you please help?
Edit: Perhaps the problem lies with proving that the class $\mathcal L$ of $E \in \mathcal X \times \mathcal Y$ for which $\nu[y:(x,y) \in E]$ is measurable is a $\lambda$-system. It is required that, if $A \in \mathcal L$, $A^C \in \mathcal L$, i.e., $\nu[y:(x,y) \in A^C]$ is measurable $\mathcal X$. I would try to prove measurability by showing that, for $c\in \mathbb R$, $\{x: \nu[y:(x,y) \in A^C]\leq c\} \in \mathcal X$. To do that, I would use: $\nu[y:(x,y) \in A^C] = \nu[Y] - \nu[y:(x,y) \in A]$ (1) and then rearrange the inequality that determines the set. However, if the measure is not finite, (1) is not lecit. I am not sure, however, if this is a real problem or there is a different way to prove measurability that I just can't see. Can you confirm/suggest anything?