The F. and M. Riesz theorem

Given: Countable choice and a finite complex Borel measure $\mu$ on the circle with $\widehat{\mu }(n)=0$ for every n<0; put $f=P[\mu ]$.

The analytic Poisson coefficient criterion gives f holomorphic and in H1, with $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}\le \mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$. Under CC complex circle measures have finite regular total variation. (Analytic Poisson integrals are exactly the measures with vanishing negative coefficients, Complex circle measures have finite regular total variation under countable choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

The CC analytic Hardy boundary theorem gives $f^{\ast }\in L^1$, finite nontangential limits, $\mathrm{\parallel }{f}_r-f^{\ast }{\mathrm{\parallel }}_1\to 0$ and $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_1=\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}$. (Fatou's boundary theorem for analytic Hardy spaces, Analytic Hardy spaces on the unit disc)

The circle Fourier coefficients of $(P[\mu ]{)}_r$ are $r^{\mathrm{\mid }n\mathrm{\mid }}\widehat{\mu }(n)$, and equality of all Fourier coefficients determines a complex circle measure under CC. Fourier coefficients of L1 functions are bounded linear integrals. (Finite complex circle measures are determined by Fourier coefficients and Poisson integrals, Fourier coefficients and trigonometric polynomials on the torus)

An L1 density h defines a complex measure h m by dominated convergence. Its known finite positive variation is supplied by the local CC circle-variation lemma; the direct unit-bounded simple-integral and phase-approximation argument gives $\mathrm{\mid }hm\mathrm{\mid }(E)={\int }_E\mathrm{\mid }h\mathrm{\mid }dm$. (A complex L^1 density defines a complex measure whose total variation is |h| dmu, Complex circle measures have finite regular total variation under countable choice)