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JCTA, Volume 151, 2017, pp. 84–101. We provide a complete proof of the Ahlswede–Khachatrian theorem in the μ p setting: for all values of n, t and p, we determine the maximum μ p-measure of a t-intersecting family on n points, and describe all optimal families (except for a few exceptional parameter settings). Our proof is based on several different articles of Ahlswede and Khachatrian. The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings. One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. 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