Tonelli’s Theorem

• categories: Measure Theory, Real Analysis, Theorem

• Statement: Let $f:{\mathbb{R}}^{n+m}\to [0,\infty ]$ be a non-negative measurable function on the product space ${\mathbb{R}}^n\times {\mathbb{R}}^m$. Then:

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

• Intuition: Tonelli’s theorem allows us to interchange the order of integration for non-negative functions without requiring that the function is integrable. This is a key generalization of Fubini’s Theorem, applicable to non-negative functions that may not be in $L^1({\mathbb{R}}^{n+m})$.

• Key Conditions:

  1. Non-negativity: The function $f(x,y)\ge 0$ must be non-negative.
  2. Measurability: $f$ must be measurable on the product space ${\mathbb{R}}^{n+m}$.

• Applications: Tonelli’s theorem is especially useful when dealing with non-negative functions that are not necessarily integrable. It is frequently used in probability theory, measure theory, and in the derivation of more general integration theorems like Fubini’s theorem.

• Difference from Fubini’s Theorem: WhileFubini’s Theorem requires $f\in L^1({\mathbb{R}}^{n+m})$, Tonelli’s theorem applies to any non-negative measurable function, even if $f$ is not integrable.