Simple functions and the integral

A nonnegative simple function takes only finitely many nonnegative values on measurable sets. In its canonical representation, those sets are disjoint level sets, so its integral is the sum of value times measure. The convention zero times infinity equals zero is used for a zero level on an infinite-measure set. Refining two descriptions to a common partition proves that the answer does not depend on the representation. The function itself, rather than a particular written sum of indicators, is what gets integrated.

An expression such as two times the indicator of A plus three times the indicator of B need not use disjoint sets. On the overlap the actual height is five. One may refine into A outside B, B outside A, and the intersection, or use the already established linearity of simple-function integration. Counting the overlap only once with height two or three would be wrong. The ability to split measurable regions without changing the answer is the concrete role played by measure additivity.

For a nonnegative measurable f, define its integral as the supremum of the integrals of nonnegative simple functions lying below it. The result may be infinite. A useful explicit approximation first truncates f at height n, then rounds downward to the nearest multiple of two to the minus n. This produces finitely many levels and measurable level sets. Because both the cutoff and resolution improve, these simple functions increase pointwise to f, including at points where f is infinite.

The approximation organizes the range into value bands, rather than requiring f to behave smoothly across intervals in the domain. That distinction handles functions such as the indicator of the rationals, which has no ordinary Riemann integral on [0,1] but has Lebesgue integral zero. The definition also yields monotonicity immediately: every lower simple approximation to f is eligible for a larger function g when f≤g. Proving the full limit and linearity theorems requires further arguments, which the next lessons develop.

For a real measurable function, split it into the positive part and negative part. Both are nonnegative, and their sum is its absolute value. The finite Lebesgue integral is defined when that absolute value has finite integral; then both part integrals are finite and can safely be subtracted. One can also define an extended integral when exactly one part is infinite. If both are infinite, the expression infinity minus infinity is undefined. Oscillation or symmetry does not repair that failure within ordinary Lebesgue integration.

Null modifications leave integrals unchanged: if integrable measurable functions agree almost everywhere, their positive and negative parts have the same respective integrals. On an incomplete measure space, however, changing values on an arbitrary subset of a null set may destroy measurability, so that qualification is not decorative. Once functions are integrable, linearity and the triangle inequality follow, and finite integrals behave like the familiar area calculations. The construction explains both their strength and the exact assumptions behind their algebra.