linearity of the integral

The integral plays nicely with the two operations that build new functions out of old ones: adding functions and scaling them. Integrating a sum gives the sum of the integrals, and scaling a function scales its integral by the same factor. In a phrase, the integral is a linear map — it commutes with the algebra of functions, just as you would hope an averaging or accumulating operation should.

Precisely, if f and g are integrable and a, b are scalars, then af + bg is integrable and the integral of (af + bg) equals a times the integral of f plus b times the integral of g. Equivalently, the set of integrable functions is a vector space and the integral is a linear functional on it. For complex scalars this includes pulling i through the integral.

Linearity sounds obvious but its proof is layered: easy and direct for simple functions, then extended to non-negative functions via monotone convergence, then to signed and complex functions through the positive/negative split. A subtle point: additivity of the integral for non-negative functions, the integral of f+g equals the integral of f plus the integral of g, must be proven before the general case and itself leans on the convergence machinery.