Fubini–Tonelli theorem

The Fubini–Tonelli theorem is the license to compute a double integral as two single integrals, in either order. Intuitively, the volume under a surface can be found by sweeping out the area of vertical slices and adding them up, and it should not matter whether you slice front-to-back or left-to-right. The theorem says exactly when this intuition is rigorous — and, importantly, when it can fail.

It comes in two halves on sigma-finite product spaces. Tonelli's half: if f is non-negative and measurable, then the double integral equals each iterated integral, with no integrability assumed in advance — the answer is the same (possibly +infinity) whichever way you proceed. Fubini's half: if f is integrable on the product (the integral of |f| is finite), then again all three integrals agree and are finite, and the inner integrals are finite for almost every fixed value of the other variable. In practice one runs Tonelli on |f| first to confirm integrability, then applies Fubini to f itself.

The hypotheses are not decoration. If f changes sign and is not absolutely integrable, the two iterated integrals can genuinely disagree. A standard cautionary example on the unit square uses f(x, y) = (x^2 - y^2)/(x^2 + y^2)^2: integrating in one order gives pi/4 and the other order gives -pi/4. The double integral of |f| is infinite, so neither Fubini nor Tonelli applies, and order suddenly matters.