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Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be sigma-finite measure spaces, and let $f$ be measurable for the product sigma-algebra. Ifthen its sections are integrable for almost every fixed coordinate, the resulting section integrals are integrable, andValues on the exceptional null sets can be assigned arbitrarily. The complex-valued version follows by real and imaginary parts.^[ref]

A useful order of reasoning is to apply Tonelli theorem to $|f|$ first. Once that nonnegative double integral is finite, Fubini justifies either signed order. For $f(x,y)=xy$ on $[0,1{]}^2$, absolute integrability is immediate and either order gives $1/4$. Merely obtaining two finite iterated signed integrals does not prove joint integrability; finite iterated integrals without joint integrability gives an explicit counterexample. The absolute-value hypothesis prevents hidden cancellation of infinite positive and negative contributions.

For measurable functions $f,g$ with values in the real numbers or complex numbers on the same measure space $(X,\mathcal{A},\mu )$, and $1\le p\le \infty$, membership in the same Lp space givesThus adding two functions cannot make their combined Lp norm exceed the sum of their separate Lp norms. The measure need not be finite or a probability measure. The hypotheses require finite Lp norms; $p=\infty$ uses the essential supremum of the absolute value, not an ordinary pointwise supremum. For complex numbers, the absolute value is the complex modulus. As usual, Lp spaces identify functions equal almost everywhere. ^[ref]

The Minkowski inequality makes the Lp norm a workable notion of size: it supplies the triangle inequality needed to turn $d_p(f,g)=\Vert f-g{\Vert }_p$ into a metric on Lp space. Applied to $(f-h)+(h-g)$, it says that travelling through an intermediate function $h$ cannot shorten the direct distance measured by this metric. Applied to two error functions, it gives a bound on the total error even when the errors reinforce one another; Minkowski inequality for error aggregation gives explicit examples.

The geometry is already visible in the Euclidean plane. Under counting measure on two coordinates, the closed unit balls for the L1 norm, Euclidean norm, and L-infinity norm are respectively $|x|+|y|\le 1$, $x^2+y^2\le 1$, and $\max (|x|,|y|)\le 1$. Every one is a convex set: if $u,v$ have Lp norm at most one, the Minkowski inequality and absolute homogeneity of a norm give $\Vert tu+(1-t)v{\Vert }_p\le t+(1-t)=1$ for $0\le t\le 1$. The Euclidean norm has no flat boundary segments, whereas the other two norms do. This is the distinction captured by a strictly convex normed space and developed in equality in Minkowski inequality.

A short Hölder's inequality proof outline handles $1<p<\infty$. First $|f+g{|}^p\le 2^{p-1}(|f{|}^p+|g{|}^p)$, so $f+g$ has finite Lp norm before any division. With conjugate exponents $p$ and $q=p/(p-1)$, set $h=|f+g{|}^{p-1}$; then $\Vert h{\Vert }_q=\Vert f+g{\Vert }_p^{p-1}$. The scalar triangle inequality followed by Hölder's inequality givesDivide if $\Vert f+g{\Vert }_p>0$; otherwise the desired bound is immediate. For $p=1$, integrate $|f+g|\le |f|+|g|$. For $p=\infty$, the almost everywhere bounds $|f|\le \Vert f{\Vert }_{\infty }$ and $|g|\le \Vert g{\Vert }_{\infty }$ give the result by the essential supremum. The same scalar triangle inequality works for real numbers and complex numbers.

Sheldon Axler authored texts on real analysis and linear algebra. The cited measure-and-integration text develops a framework in which limits and integrals can be compared through explicit convergence hypotheses, rather than relying on formal manipulation alone.^[ref]

For sigma-finite measure spaces $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$, a nonnegative product-measurable function satisfieswith $+\infty$ permitted. Nonnegativity removes cancellation, so finiteness need not be established first. The theorem starts with rectangles, extends the measurable-set identity by a Monotone class theorem, and then uses nonnegative simple functions and the monotone convergence theorem.^[ref]

For $f(x,y)=e^{-x-y}$ on $[0,\infty {)}^2$, either order gives one. Applying Tonelli to $|g|$ can establish the absolute integrability needed for Fubini theorem. The sigma-finite hypothesis is part of this standard statement: with Lebesgue measure in one coordinate and uncountable counting measure in the other, the diagonal indicator on $[0,1{]}^2$ has iterated integrals one and zero. Nonnegativity alone does not make every arbitrary pair of measures interchangeable.