Basic Analysis I — Real Analysis · Problem 6.2.22

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-improper-integrals

Problem

(requires \S5.5): Show that uniform convergence is not enough to pass the limit through improper integrals over infinite intervals. That is, find a sequence of functions fn:R→Rf_n : \mathbb{R} \to \mathbb{R} Riemann integrable on every bounded interval, converging uniformly to zero, and such that ∫−∞∞fn=1\int_{-\infty}^{\infty} f_n = 1 for every nn.
Intuitive explanation

The key insight is that uniform convergence controls pointwise behavior, but an improper integral over R\mathbb{R} also depends on how mass is distributed along the domain—you can concentrate all the area into narrower and narrower regions while staying uniformly close to zero everywhere. Think of a sequence of tall, thin bumps that all have the same total area but shrink in width and grow in height; they converge uniformly to zero because their maximum height vanishes, yet each integral remains 1.

Intuitive explanation

The key insight is that uniform convergence controls pointwise behavior, but an improper integral over R\mathbb{R} also depends on how mass is distributed along the domain—you can concentrate all the area into narrower and narrower regions while staying uniformly close to zero everywhere. Think of a sequence of tall, thin bumps that all have the same total area but shrink in width and grow in height; they converge uniformly to zero because their maximum height vanishes, yet each integral remains 1.

Related theory

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