Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?

Question

(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)

Imagine an introductory probability course that assumes the students have had first-year calculus and understand mathematical reasoning. At some point in such a course has explicated several families of discrete probability distributions, including the binomial distribution as the number of "successes" needed to get a specified non-random number of trials, and the negative binomial distribution as the number of failures before a specified non-random number of successes. Then one develops the Poisson distribution as a limit of binomial distributions with the expected number of successes remaining constant as the number of trials grows and the probability of success on each trial decreases, being inversely proportional to the number of trials.

One reaches a point in the exposition where different Poisson-distributed random variables correspond to different subsets of the line, the expected value being $\mu$ times the measure of the subset, where $\mu$ is the same for all such subsets, and these random variables are independent when their corresponding subsets are essentially disjoint, and otherwise dependent and positively correlated.

At this point all probability distributions one has dealt with are discrete.

It seems that in conventional textbooks, one does not go from there do dealing with particular continuous probability distributions until one has dealt with continuous distributions in general, stating that they have continuous cumulative distribution functions, and in absolutely continuous cases, are characterized by a density function, the value of whose integral over a set is the probability assigned to that set.

But alternatively, suppose one has not yet done what is described in the paragraph above, but one has done what is described in the paragraph before that.

One can then let $T$ be the time until the first Poisson arrival after time $0$ and let $N_t$ be the number of arrivals before time $t,$ and observe this: $$ \Pr(T>t) = {} \, \underbrace{\Pr( N_t=0) = \frac{(\mu t)^0 e^{-\mu t}}{0!}}_\text{This part has already been established.} \,= e^{-\mu t}. $$ From there, it is easy to show that $$ \Pr(T\le t) = \int_0^t e^{-\mu u} \big( \mu\, du\big). $$

Then one is dealing with a continuous distribution without having done what I reported to be done in conventional textbooks.

This has pedagogical advantages, upon which I may remark in comments if anybody cares.

My question is whether there is some published textbook that makes this kind of transition from discrete to continuous without first treating continuous distributions in general?

Answer 1

Such a transition from the discrete to the continuous is precisely the point of Nelson's Radically Elementary Probability Theory (REPT). It recently turned out that when viewed as a subsystem of BST, REPT is conservative over ZF+ADC. The relevant references are the following:

Nelson, Edward. Radically elementary probability theory. Annals of Mathematics Studies, 117. Princeton University Press, Princeton, NJ, 1987.

Nelson's book is too brief to serve as a textbook, but Herzberg published a detailed follow-up that can:

Herzberg, Frederik S. Stochastic calculus with infinitesimals. Lecture Notes in Mathematics, 2067. Springer, Heidelberg, 2013.

The book by Herzberg makes the approach accessible to students untrained in measure theory, and can be used as an undergraduate textbook.

Furthermore, detailed presentations of measure and probability theory as well as stochastic analysis from the viewpoint of nonstandard analysis appear in the book edited by Loeb and Wolff:

Nonstandard analysis for the working mathematician. Second edition. Edited by Peter A. Loeb and Manfred P. H. Wolff. Springer, Dordrecht, 2015. xv+481 pp.