Moving on to the next chapter “Random Variables – I”, take a look at the following problem. Show that the random variable is continuous if and only if for all .
(Forward direction) Suppose is a continuous random variable, then its distribution function is also continuous by definition. Hence, by definition of continuity.
(Reverse direction) Suppose for all and let be the corresponding distribution function. Let be a sequence of sets such that such that . Then, because is countably additive. Thus, and since we see that for any which is the definition of continuity.