The investigation is initially a continuation of Neuberger's work on linear boundary value problems. A very general iterative procedure for solution of these problems is described. The alternating-projection theorem of von Neumann is the mathematical starting point for this study. Later theorems demonstrate the validity of numerical approximation for Neuberger's method under certain conditions. A sampling of differential equations within the scope of our iterative method is given. The numerical evidence is that the procedure works well on neutral-state equations, for which no software is written now.

This dissertation is divided into two parts. The first part addresses the following problem: Suppose 𝑣 is a finitely additive probability measure defined on the power set 𝒜 of the integer Z so that each singleton set gets measure zero. Let X be a product space Π/β∈B * Zᵦ where each Zₐ is a copy of the integers. Let 𝒜ᴮ be the algebra of subsets of X generated by the subproducts Π/β∈B * Cᵦ where for all but finitely many β, Cᵦ = Zᵦ. Let 𝑣_B denote the product measure on 𝒜ᴮ which has each factor measure a copy of 𝑣. A subset E of X is said to be 𝑣_B -measurable iff [sic] there is only one finitely additive probability on the algebra generated by 𝒜ᴮ ∪ [E] which extends 𝑣_B. The set E ⊆ X is said to be universally product measurable (u.p.m.) iff [sic] for each finitely additive probability measure μ on 𝒜 which gives each singleton measure zero,E is μ_B -measurable. Two theorems are proved along with generalizations. The second part of this dissertation gives a proof of the following theorem and some generalizations: There are 2ᶜ nonisomorphic subalgebras of the power set algebra of the …