A new definition of conditional expectation for finite uncertainty spaces

This paper continues the authors’ previous work on developing a theory of conditional expectations in uncertainty spaces. In a previous paper, they adopted the standard definition from classical probability by defining the conditional expectation E[X|𝒢] of an uncertain variable X with respect to a σ-algebra 𝒢 as a 𝒢-measurable function provided by a version of the Radon-Nikodym Theorem for uncertainty spaces. In this current work, a definition is provided by minimizing the expected mean squared error (X − Y)² among 𝒢-measurable functions Y. The development, adopted from an existing work on non-additive probability spaces and repurposed for the current setting, similarly assumes a finite sample space and hence finitely many atoms for 𝒢. It also justifies the existence of conditional expectations and discusses some of their properties.