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 ...

Get Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.