On completeness of orthogonal systems and Dirac deltas

Given a positive measure μ supported on a set Ω ⊆ C, an orthonormal system {ϕn}n ⩾ 0 and a point a ∈ Ω, we study the relationship among μ({a}), the kernels Kn(a, a) = ∑k = 0nϕk(a)ϕk(a) and the denseness of span {ϕn}n ⩾ 0 in L2(μ) and in L2(v), where v = μ + Mδa.