What does the notation $\sum_i (A_i \times B_i)$ mean?

Question

I am reading some notes on measure theory and I came across the notation: $\sum_i (A_i \times B_i)$.

Here $A_i$ is a subset of a set $X$ and $B_i$ is a subset of a set $Y$.

What is meant by this notation?

Answer 1

I would guess that it means the set \begin{align*} \sum_i (A_i \times B_i) = \{(a,b) : a = \sum_i a_i, b = \sum_i b_i \text{ where }a_i \in A_i, b_i \in B_i\} \end{align*} i.e. the set of ordered pairs that can be written as a sum where the $i^{th}$ term in the sum is an element in $A_i \times B_i$.