The book "A Primer of Lebesgue Integration" by H.S. Bear defines Lebesgue integration through lower and upper sums $L(f,P) = \sum m_i\mu(E_i)$ and $U(f,P)=\sum M_i\mu(E_i)$ where infinite countable partitions are allowed.
The typical definition of Lebesgue integration that one encounters involves the supremum of simple functions. These simple functions are like lower sums. However, they are finite linear combinations and not infinite linear combinations.
When approaching Lebesgue integration through upper and lower sums why is it necessary to consider infinite countable partitions when simply functions seem to deal with only finite partitions?