A calculus proof of the Cram\'er-Wold theorem

TL;DR

This paper provides a simple, elementary proof of the Cramér-Wold theorem, demonstrating that a probability measure is uniquely determined by its values on half-spaces without using Fourier transforms.

Contribution

It introduces a novel, Fourier-transform-free proof of the Cramér-Wold theorem, simplifying understanding and teaching of this fundamental result.

Findings

Proof is shorter and more elementary than previous proofs

Avoids Fourier transform techniques, making the proof more accessible

Confirms the uniqueness of probability measures via half-space evaluations

Abstract

We present a short, elementary proof not involving Fourier transforms of the theorem of Cram\'er and Wold that a Borel probability measure is determined by its values on half-spaces.