Lagmental Vicfred

Fatou's Lemma Gives the Safe Direction for liminf by Vicfred

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

For nonnegative functions, the integral of a lower limit is at most the lower limit of the integrals. This is a compact note, but the quantifiers and hypotheses stay on the page.

Notation

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.

$$ f_n\ge0\ \text{measurable} $$

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 \(\mathsf C\).

$$ \int_X\liminf_{n\to\infty}f_n\,d\mu\le\liminf_{n\to\infty}\int_Xf_n\,d\mu $$

Stress the formula

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

$$ g_n=\inf_{k\ge n}f_k\uparrow\liminf_kf_k,\qquad\int g_n\le\inf_{k\ge n}\int f_k $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_n\ge0\ \text{measurable},\\[5pt] \mathsf{C}\;&:\quad \int_X\liminf_{n\to\infty}f_n\,d\mu\le\liminf_{n\to\infty}\int_Xf_n\,d\mu. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_X\liminf_{n\to\infty}f_n\,d\mu\le\liminf_{n\to\infty}\int_Xf_n\,d\mu \end{gathered}} $$

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 Wed 27 April 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.