Measure theory is concerned with assigning a “size” called a measure to subsets of a sample space Ω. Depending on the problem, this size might be a length, an area, or a volume in Euclidean space; this is known as the Lebesgue measure which is defined below. In our case, we are interested in assigning a probability measure to subsets of Ω that comprise the σ-field . For uncountable experiments, it is not possible to assign probabilities in a consistent manner to . The power set of is too large (it has cardinality beth two ), so instead we choose to be , which is the Borel σ-field on the real line and has cardinality beth one (the same as ). The Borel σ-field is useful for ...

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