The work connects the existence of a measure from analysis and probability with filters and cardinal numbers from set theory. It first introduces the properties of well-orderings, ordinals, and cardinals, and introduces the concept of transfinite induction and recursion. It then proves the equivalence between the existence of a measure and the existence of an ultrafilter which leads to the introduction of measurable cardinals. Finally, it proves the inaccessibility of measurable cardinals and connects them to trees and partitions from combinatorics.
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