Simpson's Paradox Revisited: A Measure-Theoretic and Systemic Perspective

This work revisits Simpson’s paradox from a measure-theoretic and structural perspective, demonstrating that the apparent reversal of conditional probabilities arises not from statistical surprise but from comparing quantities defined on non-invariant probability spaces. Building on measure theory, aggregation operators, and the concept of moving probability spaces, the paper shows that Simpson-type reversals are mathematically inevitable whenever aggregation alters the underlying σ-algebra or measurable structure. The manuscript formalizes the general impossibility of preserving conditional probabilities across heterogeneous groups, clarifies the role of σ-algebra inclusion, and provides a unified explanation that complements—rather than contradicts—causal accounts such as Pearl’s. Classical examples are reconstructed using explicit probability spaces, and the broader phenomenon is connected to non-closure principles observed in economics, evaluation design, and systemic analysis. Overall, the paper reframes Simpson’s paradox as a structural inconsistency caused by measurable-space asymmetry, offering a rigorous foundation for understanding aggregation, heterogeneity, and the limits of statistical inference.