Lagmental Vicfred

Tonelli Integrates Nonnegative Functions in Either Order by Vicfred

tags: mathematics, measure theory, Lebesgue integration, convergence theorems

Nonnegative measurable functions may be iterated even when the common integral is infinite. The point is to make the formal expression readable enough to audit line by line.

Statement

Lebesgue integration treats a measurable function \(f:X\to[-\infty,\infty]\) through level sets and simple approximations. The space \(L^1(\mu)\) consists of integrable functions modulo equality almost everywhere.

$$ f:X\times Y\to[0,\infty]\ \text{measurable} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \int_{X\times Y}f\,d(\mu\times\nu)=\int_X\!\left(\int_Yf(x,y)\,d\nu(y)\right)d\mu(x) $$

Worked algebra

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \int_X\int_Yf\,d\nu\,d\mu=\int_Y\int_Xf\,d\mu\,d\nu\in[0,\infty] $$

Conceptual compression

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f:X\times Y\to[0,\infty]\ \text{measurable},\\[5pt] \mathsf{C}\;&:\quad \int_{X\times Y}f\,d(\mu\times\nu)=\int_X\!\left(\int_Yf(x,y)\,d\nu(y)\right)d\mu(x). \end{aligned} $$

Caveat

Every convergence theorem has a different hypothesis. Pointwise convergence alone does not permit an integral and a limit to change places.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_{X\times Y}f\,d(\mu\times\nu)=\int_X\!\left(\int_Yf(x,y)\,d\nu(y)\right)d\mu(x) \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Mon 14 January 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.