integral of a simple function

A simple function takes only finitely many values, each on its own measurable set — think of a staircase whose steps can sit over arbitrarily weird floor-plans, not just intervals. Integrating it is the most natural thing imaginable: for each value, multiply that value by how much floor it sits over (the measure of the set), then add up. The integral is literally a weighted sum of values, weights being measures.

Precisely, if a non-negative simple function is written in canonical form as the sum of c_k times the indicator of a measurable set A_k, where the c_k are the distinct values and the A_k are disjoint, its integral is the sum of c_k times μ(A_k). One checks this is well-defined: any other representation as a combination of indicators gives the same number, so the definition does not depend on how you wrote the function. By convention 0 times infinity is taken as 0 here.

This is the seed from which the whole Lebesgue integral grows: every non-negative measurable function is a supremum of simple functions below it, and the integral of such a function is the supremum of the simple integrals. So getting this base case right — and proving it is additive and monotone — does most of the early work.