##  [Fubini's Theorem](/fubinis-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.