We trace how measure theory unifies area, mass, and probability, and why three simple rules—empty set has zero, non-negativity, and countable additivity—hold the whole framework together. We’ll unpack monotonicity, continuity from above with its finiteness caveat, and classic infinite-set counterexamples. Then we glimpse into signed measures, finite additivity, and the strange consequences of the axiom of choice (like Vitali sets), which reveal why some subsets simply cannot be assigned a size. A compact tour of the foundations that tame infinity.

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