Wikipedia Page for the Poisson Kernel

Question

On the Wikipedia page for the Poisson kernel, it gives the following formula: $$u(re^{i\theta})=\int_{-\pi}^\pi\frac{1-r^2}{1-2r\cos(\theta)+r^2}f(e^{i\theta})d\theta$$ where $u$ is holomorphic on the open unit disc, $f$ is $L^1$ on the boundary of the unit disc, and $u$ corresponds with $f$ almost everywhere on the boundary of the disc. Does this mean that, given a general measure $\mu,$ and $f\in L^1(\partial B_1(0),\mu),$ then $u$ corresponds with $f$ $\mu$ almost everywhere with the following formula $$u(re^{i\theta})=\int_{-\pi}^\pi\frac{1-r^2}{1-2r\cos(\theta)+r^2}f(e^{i\theta})d\mu(\theta)$$ or do we lose claims about u and f when we do this? If so, is there a list of conditions $\mu$ must satisfy so that nothing changes?