L1 convolution bound ($\Vert g\ast h{\Vert }_1\le \Vert g{\Vert }_1\Vert h{\Vert }_1$)

Tonelli's theorem gives the product integral of $|g(x-y)h(y)|$ as $\Vert g{\Vert }_1\Vert h{\Vert }_1$. Fubini's theorem then defines the convolution almost everywhere and gives an integrable result with norm at most that product. Multiplication by a complex exponential of modulus one preserves absolute integrability and permits the usual Fourier transform convolution identity.