Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who first proved this result in the half-plane?
Theorem: A function $F(x,y)$ defined in the upper half-plane $y > 0$ is non-negative and harmonic iff $$ F(x,y) = \int_{-\infty}^{\infty}\frac{y}{(t-x)^{2}+y^{2}}d\mu(t)+Ay,\;\; y > 0, $$ for some non-negative constant $A$ and a non-decreasing real function $\mu$ on $\mathbb{R}$ for which $$ \int_{-\infty}^{\infty}\frac{1}{1+t^{2}}d\mu(t) < \infty. $$ The function $\mu$ is unique if $\mu$ is normalized so that $\lim_{y\downarrow x}\mu(y)=\mu(x)$ and $\mu(0)=0$.
Note: This representation theorem may be stated in terms of a positive Borel measure $\mu$ instead; in that case $\mu$ is unique as a Borel measure.
This result is Theorem 4 in
Loomis, L. H.; Widder, D. V. The Poisson integral representation of functions which are positive and harmonic in a half-plane. Duke Math. J. 9, (1942). 643–645.
As explained in MathSciNet, a main contribution of the paper is that the proof does not use conformal invariance with the disk. Otherwise, the formula was obtained earlier by Verblunsky (Proc Cambridge Math Soc, 1935).
An n-dimensional version is also true, and a proof along the same lines can be seen in
Rudin, Walter Tauberian theorems for positive harmonic functions. Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 3, 376–384.