I've got an assignment to consider boundary problems in space of finite measures W, where the boundary conditions are considered in sense of weak approximation.
For example Hilbert's problem with boundary condition in sense of weak approx. will have following definition. $$ \lim_{r \to 1-0}\int\limits_T \ ({\Phi^+(rt)}-a(t){\Phi^-(r^{-1}t)})g(t)dt=\int\limits_Tg(t)d\mu $$ Where $$ T = \{ z: |z|=1 \},\ \forall g \in C,\ d\mu \in W \\\Phi^+(z)\ - \text{analytic function inside unit circle}\\ \Phi^-(z)\ -\text{analytical function outside the unit circle}\\ a(t) - \text{is a piecewise continuous function from Hölder's space.} $$
Questions :
- I would like to understand what is the idea of formulating the problem in measure space ?
- What generalization it does ?