Q:

Lebesgue integration and countable partitions

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?

A:

After thinking about my own question I think the example $f=1/x^2$ highlights the problem with using simple functions for upper sum approximations. There is no problem in using simple functions from below to approximate $\int_1^\infty 1/x^2dx$ from below. But a simple function $\phi\ge f$ approximating from above will necessarily have infinite integral because $\phi$ will not be able to take on arbitrarily small values to well approximate $1/x^2$ for large $x$.

