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.