Fubini’s Theorem

• categories: Measure Theory, Real Analysis, Theorem

• Statement: Let $f:{\mathbb{R}}^{n+m}\to \mathbb{R}$ be a measurable function on the product space ${\mathbb{R}}^n\times {\mathbb{R}}^m$.

  1. If $f\in L^1({\mathbb{R}}^{n+m})$, then:

     $${\int }_{{\mathbb{R}}^n}\left({\int }_{{\mathbb{R}}^m}f(x,y)dy\right)dx={\int }_{{\mathbb{R}}^m}\left({\int }_{{\mathbb{R}}^n}f(x,y)dx\right)dy={\int }_{{\mathbb{R}}^{n+m}}f(x,y)d(x,y)$$

     In other words, the order of integration can be interchanged.

  2. For non-negative measurable functions $f\ge 0$, the integrals may not be finite, but the result still holds:

     $${\int }_{{\mathbb{R}}^n}\left({\int }_{{\mathbb{R}}^m}f(x,y)dy\right)dx={\int }_{{\mathbb{R}}^{n+m}}f(x,y)d(x,y)$$

• Intuition: Fubini’s theorem allows us to compute double integrals by integrating iteratively, first over one variable and then over the other. It ensures that for functions that are either integrable or non-negative, the order of integration can be swapped without affecting the result.

• Key Conditions:

  1. Integrability: $f\in L^1({\mathbb{R}}^{n+m})$ guarantees that the integrals are finite and the theorem applies.
  2. Non-negative functions: For non-negative measurable functions, even if the function is not integrable, the double integral can still be computed iteratively.

• Applications: Fubini’s theorem is used in various contexts, such as probability theory, multivariable calculus, and measure theory, where changing the order of integration simplifies calculations.