Lagmental Vicfred

Monotone Convergence Interchanges an Increasing Limit and Integral by Vicfred

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

Nonnegative measurable functions increasing pointwise may pass their limit directly through the integral. A small computation will anchor the general statement before the abstraction takes over.

Set-up

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.

$$ 0\le f_1\le f_2\le\cdots,\qquad f_n\uparrow f $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \int_Xf\,d\mu=\lim_{n\to\infty}\int_Xf_n\,d\mu $$

The calculation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ f_n=\sum_{k=1}^{n}a_k\mathbf1_{E_k},\ a_k\ge0\quad\Longrightarrow\quad\int\sum_{k\ge1}a_k\mathbf1_{E_k}\,d\mu=\sum_{k\ge1}a_k\mu(E_k) $$

What survives abstraction

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad 0\le f_1\le f_2\le\cdots,\qquad f_n\uparrow f,\\[5pt] \mathsf{C}\;&:\quad \int_Xf\,d\mu=\lim_{n\to\infty}\int_Xf_n\,d\mu. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_Xf\,d\mu=\lim_{n\to\infty}\int_Xf_n\,d\mu \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Wed 21 October 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.