ABSTRACT
Calculus and set theory are the tools for adapting what we have done so far to the more complicated continuous setting. Remember that when the sample space of a random experiment is finite or countable, to construct a probability it is enough to make an assignment P[Ω] of positive probability to each outcome Ω . To be valid, these outcome probabilities must be non-negative and must sum to 1. In continuous probability, we are usually interested in sample spaces which are intervals. But the problem is that we cannot assign strictly positive probability to all points in an interval a, b without violating the condition that the sample space has probability 1. To see this, observe that in order for P a, b to be less than or equal to 1, we can have at most 1 point in a, b of probability in 1 2, 1, at most 2 points of probability in 1 3, 1 2, at most 3 points of probability in 1 4, 1 3, etc. The set of points in a, b of strictly positive probability is the disjoint union as n ranges from 2 to of the sets of points whose probability is in 1 n, 1 n 1. Since the set of points of positive probability is therefore a countable union of finite sets, it is at most countable. In particular it follows that the set of points of positive probability on can contain no intervals.
