Lagmental Vicfred

Dominated Convergence Uses One Integrable Envelope by Vicfred

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

Pointwise almost-everywhere convergence and domination by an L1 function justify exchanging limit and integral. I will separate the object being defined from the consequence being claimed.

Definitions first

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.

$$ f_n\to f\ \text{a.e.},\qquad|f_n|\le g\in L^1 $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \int f_n\,d\mu\longrightarrow\int f\,d\mu $$

A small case in full

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \left|f_n-f\right|\le2g,\qquad\int|f_n-f|\,d\mu\to0,\qquad\left|\int f_n-\int f\right|\le\int|f_n-f| $$

The reusable statement

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_n\to f\ \text{a.e.},\qquad|f_n|\le g\in L^1,\\[5pt] \mathsf{C}\;&:\quad \int f_n\,d\mu\longrightarrow\int f\,d\mu. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int f_n\,d\mu\longrightarrow\int f\,d\mu \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 17 September 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.