Herglotz representation theorem

Remarks

The integrand in part 1 is the Poisson kernel. For $0 < r < 1$, the measures $\mu_r = u(r\zeta)\, \frac{\abs{d\zeta}}{2\pi}$ on $\mathbb{T}$ are positive with total mass $u(0)$, by the mean value property. The Poisson formula on the disc of radius $r$ expresses $u(rz)$ through $\mu_r$; a weak-star limit of the $\mu_r$ as $r \to 1$ gives $\mu$. Part 2 follows from part 1 as in the Schwarz integral formula. Example: for the Dirac measure at $1$, the formula of part 1 gives $u(z) = (1 - \abs{z}^2)/\abs{1 - z}^2$, a positive harmonic function that is unbounded near $1$, and part 2 gives $f(z) = (1 + z)/(1 - z)$. The hypothesis $u \ge 0$ is needed. The function $u(z) = \operatorname{Re} \frac{(1 + z)^2}{(1 - z)^2}$ is harmonic but takes both signs. It is not the Poisson integral of any finite signed measure, because it grows too fast near $1$.

Going deeper

For a student

The theorem identifies the cone of nonnegative harmonic functions on the disc with the cone of finite positive measures on the circle. In the holomorphic form it describes all functions with nonnegative real part, the Carathéodory class, which is used in coefficient problems for univalent functions. It also underlies spectral representations in operator theory, such as the moment problem and the spectral theorem for unitary operators. Together with Harnack's inequality it controls the size of positive harmonic functions.