第一步：简单函数的积分
把一个非负简单函数写成标准形式：s = 对 k 求和 a_k·1_{A_k}，其中 a_k 是它的两两不同的取值，A_k = {x : s(x) = a_k} 是互不相交的可测集。简单函数的积分由唯一合理的公式定义——值乘以大小，对所有片求和。
Standard form: s = sum_{k=1..n} a_k * 1_{A_k}, a_k distinct, A_k disjoint measurable
Definition: integral of s = sum_{k=1..n} a_k * m(A_k)
(here m = Lebesgue measure; use the convention 0 * infinity = 0)
Example. On [0,5] let
s = 3 on [0,2), s = 0 on [2,4), s = 7 on [4,5]
Then A_1 = [0,2), m=2 ; A_2 = [2,4), m=2 ; A_3 = [4,5], m=1
integral of s = 3*2 + 0*2 + 7*1 = 6 + 0 + 7 = 13.第二步：用上确界处理非负可测函数
现在设 f 是任意非负可测函数（允许取值于 [0, +∞]）。用简单函数从下方逼近它，并取这种逼近中最好的。非负函数的积分定义为落在 f 下方的那些简单函数积分的上确界。
Definition (f >= 0 measurable):
integral of f = sup { integral of s : s simple, 0 <= s <= f everywhere }
The sup is over a NON-EMPTY set (s = 0 always qualifies)
and it may equal +infinity. So integral of f always EXISTS in [0, +infinity].
Approximate from below — the canonical staircase:
for level n, set s_n(x) = min( n, floor(2^n f(x)) / 2^n )
-> each s_n is simple, 0 <= s_1 <= s_2 <= ... <= f,
and s_n(x) -> f(x) at every x (increasing pointwise limit).顺带得到的性质
两条基本性质从上确界定义中立刻得出。单调性： 若 0 <= f <= g，则 integral f <= integral g，因为 f 下方的每个简单函数也都在 g 下方，故 f 的上确界是在更小的集合上取的。非负性： 总有 integral f >= 0。它们呼应着测度的单调性，并将在接下来的收敛定理中被反复使用。