The Fubini-Tonelli Theorem

You are summing a two-dimensional table of numbers — say, expected loss by (scenario, asset) pair — and you want to know: does it matter whether you sum each row first and then add the row totals, or sum each column first and then add the column totals? Usually not. But "usually" is not "always," and the Fubini-Tonelli theorem is the precise statement of exactly when swapping the order is guaranteed to give the same answer, plus a warning about the cases where it silently doesn't.

The analogy: totalling a spreadsheet two ways

Take a spreadsheet grid of numbers, rows and columns. You can total it by summing along each row, then adding the row-sums; or by summing down each column, then adding the column-sums. For a finite grid of ordinary numbers these two totals are obviously the same — addition doesn't care about order. Fubini-Tonelli extends that obvious fact to infinite grids (infinitely many rows, infinitely many columns, or continuous "rows" and "columns" as in a double integral), where the "obviously" stops being obvious: an infinite grid can have row-sums and column-sums that total to different numbers if the entries are allowed to be both positive and negative.

Writing it down

Consider a function of two variables $f(x,y)$, and the double integral over both variables. Tonelli's part handles non-negative functions: if $f(x,y) \geq 0$ everywhere, then

$$\int\!\!\int f(x,y)\, dx\, dy = \int\left(\int f(x,y)\, dy\right) dx = \int\left(\int f(x,y)\, dx\right) dy .$$

In words: for a non-negative grid, integrate however you like, in whichever order, and you always land on the same total — no extra condition needed.

Fubini's part extends this to functions that can be negative, but only under one guardrail: it applies whenever

$$\int\!\!\int |f(x,y)|\, dx\, dy < \infty .$$

In words: first check that the absolute values sum to something finite (Tonelli already guarantees this check itself doesn't depend on order, since $|f|$ is non-negative). If that finite check passes, the original signed double integral is also order-independent. If it fails — the absolute values sum to infinity — order can genuinely change the answer, or the integral may not even be well defined.

Worked example 1: joint expectation of independent losses

Two independent loss variables $X$ and $Y$, each taking values 1, 2, 3 with equal probability $1/3$. You want $E[XY]$. Row-first (condition on $X$, sum over $Y$, then over $X$): for each fixed $x$, $E[XY \mid X=x] = x \cdot E[Y] = x \times 2$ (since $E[Y] = (1+2+3)/3 = 2$). Averaging over $x = 1,2,3$: $E[X \cdot 2] = 2 \times E[X] = 2 \times 2 = 4$. Column-first (condition on $Y$ instead): by the identical argument with roles swapped, $E[XY] = 2 \times E[X\text{'s mean}] = 4$ again. Both orders give $E[XY]=4$, and Fubini-Tonelli guarantees this in advance because all nine $(x,y)$ pairs are non-negative — Tonelli's part applies with no further check required.

Worked example 2: when the swap actually fails

Consider the doubly-indexed array $a_{ij} = 1$ if $i = j$, $a_{ij} = -1$ if $i = j+1$, and $a_{ij}=0$ otherwise, for $i, j = 1, 2, 3, \dots$. Sum row $i$ first, then add row totals: each row sums to $0$ (its $+1$ cancels the $-1$ from the row below it, except the pattern is arranged so each individual row nets to zero), giving a grand total of $0$. Sum column $j$ first instead: column 1 alone sums to $1$ (only the diagonal entry contributes, since there's no row above it to contribute a $-1$), and every subsequent column also nets to $0$, giving a grand total of $1$. Two different orders, two different answers: $0 \neq 1$. Check Fubini's condition: $\sum_{i,j} |a_{ij}| = \infty$, since there are infinitely many nonzero entries each of size 1. The finiteness check fails, exactly as the theorem warns, and that failure is why order mattered here.

What this means in practice

Fubini-Tonelli is the theorem quietly invoked whenever a quant reorders a nested sum or integral — computing expected P&L by summing over scenarios then assets versus assets then scenarios, or swapping the order of integration in a pricing formula involving two sources of randomness. It is also exactly the justification needed to differentiate under an integral sign, a routine step in deriving Greeks.