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Learn the formal mathematical foundations of probability theory through this 33-minute lecture that builds upon sample and event spaces to establish rigorous probability concepts. Explore indicator functions and their application in defining event occurrences and relative frequencies, then delve into the axiomatic definition of probability functions based on three fundamental axioms: non-negativity, countable additivity, and certainty. Discover key properties that emerge from these axioms, including the probability of impossible events, finite additivity, monotonicity principles, and the inclusion-exclusion principle. Master advanced concepts such as limits of sets and examine detailed proofs demonstrating the continuity of probability for both monotonic and general sequences of sets, providing essential mathematical tools for understanding probability theory at a foundational level.
