Step one: the integral of a simple function
Write a non-negative simple function in its standard form: s = sum over k of a_k·1_{A_k}, where the a_k are its distinct values and A_k = {x : s(x) = a_k} are disjoint measurable sets. The integral of a simple function is defined by the only sensible formula — value times size, summed over all the pieces.
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.Step two: non-negative measurable functions by supremum
Now let f be any non-negative measurable function (values allowed in [0, +infinity]). Approximate it from below by simple functions and take the best such approximation. The integral of a non-negative function is the supremum of the simple integrals lying under f.
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).The properties that come for free
Two basic properties fall out immediately from the supremum definition. Monotonicity: if 0 <= f <= g then integral f <= integral g, because every simple function under f is also under g, so f's sup is taken over a smaller set. Non-negativity: integral f >= 0 always. These mirror the monotonicity of measure and will be used constantly in the convergence theorems ahead.