product measures and Fubini's theorem
/ foo-BEE-nee /
How do you model two random experiments happening together — say two independent dice, or a random point in the plane? You need a single probability over all the joint outcomes (pairs), built from the two separate models. The product measure is exactly this construction: it glues two measure spaces into one space of ordered pairs in the way that makes the two coordinates independent. Fubini's theorem is the companion result that tells you a double integral over that joint space can be computed one coordinate at a time, in either order.
Given measure spaces with measures mu and nu, the product measure mu x nu on the space of pairs is defined so that a rectangle A x B gets weight (mu x nu)(A x B) = mu(A) times nu(B) — area equals width times height — and Caratheodory's extension then spreads this rule consistently to all measurable subsets of the product, not just rectangles. In probability, if X has law mu and Y has law nu, the joint law of (X, Y) being the product measure is precisely the statement that X and Y are independent. Fubini's theorem then says: for a measurable function f of two variables, the double integral equals either iterated integral — you may integrate over y first and then x, or x first and then y, and get the same answer: integral integral f d(mu x nu) = integral [ integral f dnu ] dmu = integral [ integral f dmu ] dnu.
This is the rigorous license behind swapping the order of summation or integration, and behind splitting a joint expectation of independent variables: E[g(X) h(Y)] = E[g(X)] times E[h(Y)]. The honest caveat — often forgotten — is that the swap is only guaranteed when either f is non-negative (this version is called Tonelli) or the double integral of |f| is finite (this is Fubini proper). If f changes sign and is not absolutely integrable, the two iterated integrals can give different finite answers, so you must check integrability of |f| before exchanging the order.
Two independent dice live on the product space {1,...,6} x {1,...,6} with product measure giving each pair (i, j) weight (1/6)(1/6) = 1/36. Fubini lets you compute E[X + Y] by summing over j for each i and then over i, in either order, both giving 7.
Glue two spaces so the coordinates are independent, then integrate one variable at a time in either order.
Swapping the order of integration is only safe when f is non-negative (Tonelli) or the integral of |f| is finite (Fubini); for sign-changing, non-absolutely-integrable f the two iterated integrals can genuinely disagree.