It seems like different books define continuous random variables differently.
pdf definition: The random variable X is continuous if a nonnegative function f exists, that is defined for all $x \in (-\infty, \infty)$ such that $P(X \in B) = \int_B f(x) dx$ (Sheldon Ross)
cdf definition: The random variable X is continuous if its cdf is continuous and differentiable. Moreover, the derivative of the cdf is continuous except for a finite number of points. (Wackerly)
Another cdf defintion: The random variable X is continuous if the cdf is a continuous function. (Wikipedia https://en.wikipedia.org/wiki/Random_variable#Continuous_random_variable)
uncountable range: The random variable X is continuous if its range is uncountable infinite/set of possible values is uncountable infinite.
uncountable sample space: The random variable X is continuous if the sample space is uncountable infinite.
My question are:
a) How do these definitions relate to each other? Are they equivalent? Why?
b) What is the difference between a continuous and an absolute continuous random variable?
c) Is it possible that the difference between continuous and discrete random variables vanishes from the point of view of measure theory? I have looked into several books about probability based on a rigid measure theoretic approach and none of them even mention the fact that there are different kinds of random variables.
d) In particular, is 4.) and 5.) equivalent? To me this doesnt seem to be necessarily the case...
e) Which is the "correct" definition from the above for a continuous random variable?
I would be happy also about answers from the point of view of measure theory, which probably answers this questions the most precisely.