Helson–Lowdenslager theorem

Remarks

Such an order on $\Gamma$ exists exactly when $\Gamma$ is torsion-free, that is, when $G$ is connected. For $G$ the circle and $\Gamma = \Z$ with its usual order, the theorem gives back the F. and M. Riesz theorem. Indeed $\mu_s$ is then of analytic type with $\hat\mu_s(0) = 0$, so $e^{-it}\mu_s$ is again of analytic type and singular. Repeating the argument shows that $\hat\mu_s(n) = 0$ for every $n$, so $\mu_s = 0$. On larger groups this repetition is not available, since an element of $\Gamma$ may have infinitely many elements between it and $0$. There the theorem only says how the two parts split. Helson and Lowdenslager proved it with methods from prediction theory.

Going deeper

For a student

The result belongs to the theory of $H^p$ spaces on compact groups with ordered duals. It shows that the singular part of an analytic measure is again analytic and has zero mean. On the circle this is enough to remove the singular part, because each integer has only finitely many integers between it and $0$. For the torus $\mathbb{T}^2$ with the lexicographic order on $\Z^2$, this finiteness fails, and so does the stronger conclusion. For example, the product measure $\mu = e^{ix} \, dx \otimes \delta_0$ is singular, and its Fourier–Stieltjes transform vanishes except at $(1, n)$ for $n \in \Z$, which are all positive.