Two footnotes to the F. & M. Riesz theorem

We present a new proof of the F. & M. Riesz theorem on analytic measures of the unit circle $\mathbb{T}$ that is based the following elementary inequality: If $f$ is analytic in the unit disc $\mathbb{D}$ and $0\le r\le \varrho <1$, then $$\mathrm{\parallel }{f}_r-f_{\varrho }{\mathrm{\parallel }}_1\le 2\sqrt{\mathrm{\parallel }{f}_{\varrho }{\mathrm{\parallel }}_1^2-\mathrm{\parallel }{f}_r{\mathrm{\parallel }}_1^2},$$ where $f_r(e^{i\theta })=f(r{e}^{i\theta })$ and where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ denotes the norm of $L^1(\mathbb{T})$. The proof extends to the infinite-dimensional torus ${\mathbb{T}}^{\mathrm{\infty }}$, where it clarifies the relationship between Hilbert’s criterion for $H^1({\mathbb{T}}^{\mathrm{\infty }})$ and the F. & M. Riesz theorem.