Coherent upper conditional expectations defined by fractal measures and the representation of quantum states

Coherent upper conditional expectations are introduced through fractal outer measures to consider conditioning events that have zero probability with respect to the initial probability, as fractal sets. The model can be applied when the probability is concentrated on sets with zero Lebesgue measure and the density does not exist. It can be applied to provide a probabilistic representation of quantum states in cases where the density probability is absent. However, when the density probability exists, the two probabilistic representations align. This new approach can have intriguing data analytics applications, ranging from addressing probabilistic challenges in quantum state representation in quantum physics and managing extreme events in financial markets to capturing rare occurrences with significant implications in biomedical research.