Why is the sum of two functions expressed as simple functions is the sum of the weighted indicator variable of the intersection? If
$$g=\sum_{i=1}^n a_i \mathbb I_{A_i}$$
and
$$h=\sum_{i=1}^m b_j \mathbb I_{B_j}$$
why is
$$g + h = \sum_{i=1}^n \sum_{j=1}^m (a_i + b_j) \mathbb I_{A_i\cap B_j}$$
I would have guessed that the sum would use the indicator variable $\mathbb I_{A_i \cup B_j},$ which is to say the union of both measurable sets.

Suppose $g = 3 \mathbb I_{A_1}$ and $h = 4 \mathbb I_{B_1}$.
The definition is multiplying the coefficients.  The combination has the value $12$ only in the intersection, not the union.
