A result in measure theory that gives conditions under which the integral of a measurable function on a product measure space equals the iterated integrals computed in either order; typically stated for functions that are absolutely integrable on the product space.

Principle

If f is a measurable function on X×Y with respect to the product measure and ∫_{X×Y} |f| < ∞, then for almost every x the section y ↦ f(x,y) is integrable on Y, for almost every y the section x ↦ f(x,y) is integrable on X, and ∫_{X×Y} f = ∫_X (∫_Y f(x,y) dy) dx = ∫_Y (∫_X f(x,y) dx) dy.

Demonstration

Concrete scenario: on R^2 with Lebesgue measure, take f(x,y)=sin(xy)/(1+x^2+y^2). Since ∫_{R^2} |f(x,y)| dx dy < ∞, Fubini's theorem justifies computing ∫_{R^2} f by integrating first in y and then in x (or vice versa) and obtaining the same finite value.

Misapplication

Applying Fubini when f is not absolutely integrable (∫|f| = ∞) can produce contradictory or undefined iterated integrals; for some conditionally integrable functions the two iterated integrals exist but are not equal or one diverges, so naive interchange of order is invalid.

Consequence

When its hypotheses hold one can reduce a multiple integral to iterated one-dimensional integrals, exchange order of integration freely, and apply one-dimensional tools (dominated convergence, monotone convergence) inside each iterated integral.

Reversal

Invert the claim by removing absolute integrability: Tonelli's theorem guarantees equality of iterated integrals for nonnegative measurable functions even when integrals may be infinite; without nonnegativity or absolute integrability equality can fail.

Boundary

Requires a product measure on σ-finite measure spaces in standard formulations and the absolute integrability hypothesis (∫|f|<∞); functions failing measurability, σ-finiteness, or absolute integrability lie outside the theorem's scope.

Semantic Tension

Tension arises with Tonelli's theorem (nonnegative case) and with statements about conditional integrability that allow iterated integrals to exist but differ — the boundary between absolute and conditional integrability is the key competing meaning.

Synthesis

Fubini's theorem is the tool that, under absolute integrability and standard measure-theoretic hypotheses, identifies the multiple integral with iterated one-dimensional integrals and thereby legitimizes interchange of integration order in analysis.