Consider the sample space and the partition

Let  be the field generated by this partition, and define the mapping  as follows:

This seemingly innocent mapping is not a random variable with respect to the field. In fact, we cannot assign the probability P(X=3), since

To be a random variable, the mapping should be constant for the three outcomes in {2,3,4}, which is an element of the field F. For instance, the random variable  defined as

is a legitimate random variable.

Transcribed Image Text:

= {1, 2, 3, 4}