On Planetmath, product measure is roughly defined as follows:
Let $(E_i, \mathbb{B}_i, u_i)$ be measure spaces, where $i\in I$ an index set, possibly infinite.
When each $u_i$ is totally finite, there is a unique measure on the product measurable space $(E, \mathbb{B})$ of $(E_i, \mathbb{B}_i, u_i)$. such that "taking measure" and "taking product" can be "exchanged" for any $B=\prod_{i \in I} B_i$ with $B_i \in \mathbb{B_i}$ and $B_i=E_i$ for all $i \in I$ except on a finite subset $J$ of $I$.
I was wondering if there is also a unique measure on the product measurable space, such that "taking measure" and "taking product" can be "exchanged" for any $B=\prod_{i \in I} B_i$ with $B_i \in \mathbb{B_i}$, without requiring "$B_i=E_i$ for all $i \in I$ except on a finite subset $J$ of $I$"? This is not used in the definition of product measure, and is it only because the product might be for infinite number of terms?
If $I$ is infinite, one sees that the total finiteness of $u_i$ can not be dropped. For example, if $I$ is the set of positive integers, assume $u_1(E_1) < \infty$ and $u_2(E_2)=\infty$ . Then $u(B)$ for $$B:=B_1 \times \prod_{i>1} E_i=B_1\times E_2 \times \prod_{i>2} E_i,$$ where $B_1 \in \mathbb{B}_1$ would not be well-defined (on the one hand, it is $u_1(B_1)<\infty$ , but on the other it is $u_1(B_1)u_2(E_2)=\infty$ ).
I don't understand the example. Specifically how does the last sentence in parenthesis show that the measure $u$ is not well-defined on $B$?
Thanks and regards!