On Projection Properties of Monotone Integrable Functions

This research formulates an \((i-1, i)\) - dimensional structure of \(μ|f|^p(i-1, i)\)-vector measure integrable functions for \(i=1,2, … n\). Fixed point projection properties of a vector measure are appplied to determine the measurability of sets in the domain of integrable functions. Measurable sets of the form \(Π_i Aᵢ₋₁(i, i+1)\) are partitioned into disjoint sets \(Π_i Aᵢ₋₁^i\) of finite measure.The obtained results demonstrate utility of concepts of vector measure duality, continuity from below of a measure and monotonicity of a vector measure in integrating functions.