Basic Analysis I — Real Analysis · Problem 6.2.9

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · l1-norm-not-convergent

Problem

Find a sequence {fn}n=1∞\{f_n\}_{n=1}^\infty of Riemann integrable functions on [0,1][0, 1] converging pointwise to 00, but such that lim⁡n→∞∥fn∥L1 does not exist (is ∞).\lim_{n \to \infty} \|f_n\|_{L^1} \text{ does not exist (is } \infty\text{)}.
Intuitive explanation

The key is to construct functions that shrink to zero everywhere, yet their total "mass" (the L1L^1 norm) keeps growing without bound. You'll want functions that concentrate their non-zero values on shrinking regions while getting taller to compensate—think of a sequence of narrower, taller spikes. Pointwise convergence to zero is easy to verify at each fixed point, but the integral can still diverge if the functions are engineered to preserve or increase their total area. This shows that pointwise convergence is genuinely weaker than L1L^1 convergence.

Intuitive explanation

The key is to construct functions that shrink to zero everywhere, yet their total "mass" (the L1L^1 norm) keeps growing without bound. You'll want functions that concentrate their non-zero values on shrinking regions while getting taller to compensate—think of a sequence of narrower, taller spikes. Pointwise convergence to zero is easy to verify at each fixed point, but the integral can still diverge if the functions are engineered to preserve or increase their total area. This shows that pointwise convergence is genuinely weaker than L1L^1 convergence.

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