A refinement of the Fatou-Naim-Doob theorem (Q1300180)

Let \(\Omega\) denote a Brelot harmonic space with a countable base for its topology and a positive potential. Let \(\Delta_1\) denote the minimal boundary. A set \(E\) is defined to be harmonic thin at \(u\in\Delta_1\) if there is a positive harmonic \(h\) such that \(h\geq u\) on \(E\) but \(h(x_0)<u(x_0)\) for some \(x_0\) not in \(E\). The sets \({\mathcal H}_u=\{E:\Omega\setminus E \text{ \;is\;harmonic\;thin\;at} u\}\) is seen to be a filter. The axiom HT is defined so that if \(f\) and \(g\) are positive harmonic functions whose representing measures are mutually singular, then the set where \(f>g\) is harmonic thin at \(\mu_g\)-almost every point of \(\Delta_1\). The refinement of the Fatou-Naim-Doob theorem is as follows: If the axiom HT holds, and if \(f\) and \(h\) are any two positive harmonic functions then for \(\mu_h\)-almost every \(u\) does \(f/h\) have a limit following the harmonic fine filter \({\mathcal H}_u\). In the Fatou-Naim-Doob theorem the almost everywhere convergence is along the fine filter, and the harmonic fine filter is shown to be coarser than the fine filter, hence this is an refinement of the Fatou-Naim-Doob theorem, in \textit{K. Gowrisankaran} [Ann. Inst. Fourier 16, No. 2, 455-467 (1966; Zbl 0145.15103)].