For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$ $$ \Delta u=0, \text{ on } D, \\ \left. u\right|_{\partial D}=f \in C\left( \partial D\right). $$ With a fix $x$ in $D$, an application of the Riesz representation theorem gives $$ u(x)=\int_{\partial D} f(t) d\mu_x(t), $$ for some measure $\mu_x$.
What is the operator on which Riesz theorem is applied?
Which Riesz theorem is used?