Statistical Concentration and Failure Measures under Project–Type Refinements on Atomless Probability Spaces

Abstract

In earlier work of Ili\'c and Veli\v ckovi\'c (2019) a completed project is represented by a finite set of tasks with numerical quality indices, and the overall failure of the project is measured by the largest probability assigned to any single task. 
Project revisions are modelled by replacing a problematic task with a family of improved subtasks, which, on the probabilistic level, corresponds to splitting one atom of a finite probability space into several smaller atoms.

In this paper we give a simple measure--theoretic reformulation of this refinement procedure on a general atomless probability space. 
We work with finite measurable partitions and show that suitable project–type refinements, which split all atoms of maximal probability into smaller pieces of the same total probability, strictly decrease the
project failure measure. 
Moreover, for any given threshold the failure measure can be made smaller than this threshold after finitely many such refinements, and along suitable infinite refinement sequences it converges to zero.

We then introduce a natural concentration functional which measures how strongly the total failure probability is concentrated in a few atoms, and we prove that it is also strictly decreased by the same project--type refinements and can be made arbitrarily small. 
In this way we obtain a unified probabilistic framework in which both the maximal failure probability and a basic concentration index improve under explicit local project revisions. 
The original discrete project model of Ili\'c and Veli\v ckovi\'c (2019) arises as a special case, and several small project examples are given to illustrate the effect of the refinements on concrete failure profiles.