In Axler's book "Harmonic function theory" on this link http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Livres/Harmonic%20Function%20theory.pdf on page 112 the author say: "We know that if $f\in C\left(S\right)$, then $P\left[f\right]$ has a continuous extension to $\overline{B}$. What can be said of the more general Poisson integrals defined above? We begin to answer this question in the next two theorems." I tried to understand the idea of this, but I couldn't understand. So, my first question is: "What is the idea and answer about this problem?" Then, I tried to think in other way. If we suppose that $P\left[\mu\right]$ can be extended continuously to $\overline{B}$, and if we denote by $u$ such extension, then from theorem 1.21 on page 15 we will have $u=P\left[f\right]$, where $f$ is a restriction of $u$ to $S$. Now we have that $P\left[\mu\right]=P\left[f\right]$ on $B$, and this implies that $$\int_{S}P\left(x,\xi\right)d\mu\left(\xi\right)=\int_{S}P\left(x,\xi\right)d\mu_{f}\left(\xi\right),$$ where $d\mu_{f}\left(\xi\right)=f\left(\xi\right)d\sigma\left(\xi\right)$. My intuition tells me that I need to prove that $\mu=\mu_{f}$ but I don't know how to prove that. Are there any theorems in Measure theory related to this kind of problem? So we have the same integrals of a "nice" function in two measures. What are the conditions for this "nice" function and on the set to imply the equality of these measures?