Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.15 NONMEASURABLE SUBSETS OF

As mentioned previously, the Borel σ-field is sufficient for the probability space for practical applications, and thus we use to describe events on . However, is not the only subset of ; there is in fact an infinity of subsets, of which not all are measurable and thus a probability measure cannot be assigned. Consider the interval [0, 1] for which the Borel σ-field is a subset generated by all open intervals on [0, 1]. The Lebesgue measure for any such interval (a, b) is L(a, b) = b−a for b>a, which is bounded above by 1 in this case. The Cantor set is another subset of [0, 1] that is different from the Borel σ-field and is still measurable; recall that it has Lebesgue measure zero. Nonmeasurable subsets of are generally difficult to describe. We illustrate one example using a construction leading to the Vitali set, which has a complicated form and would ...