Yes, it is the same:
A sigma algebra generated by a collection of random variables (say, real-valued) is the smallest sigma algebra that makes these random variables measurable: that means that it is the sigma algebra generated by the pre-images of any interval of the real line (with rational boundary points, if you prefer a countable generating collection).
Vice versa, on can say that a sigma algebra generated by a family of sets is the smallest sigma algebra which makes all the characteristic functions (i.e. the functions which identify these sets by being 1 inside such a set, and 0 outside) of these sets measurable.
You can find such elementary measuretheoretic concepts in any introductory textbook about measure theory. If your intention is to go deeper into math, I personally recommend the intro to measure theory by Paul Halmos, but I am sure there is plenty other great literature.