Measure, Integration & Real Analysis · Problem 11C.17

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-not-surjective

Problem

Prove there exists a continuous function

g: \mathbf{R} \to \mathbf{R}

such that

\lim_{t \to \pm\infty} g(t) = 0

and

g \notin \{\hat{f} : f \in L^1(\mathbf{R})\}