I'm not very clear on some technicalities around probability, but I remember a 3blue1brown video: something along the lines of the probability of randomly choosing an integers out of the real number line being a probability zero event; in spite of there being infinitely many integers to "randomly pick"
Try sampling random floats in your programming language of choice and see how long it takes to get an integer (you will eventually get one).
Then consider that floating point numbers represent only a finite subset of the (countably) infinite rational numbers. And then consider that the set of rational numbers is unimaginably smaller than the set of irrational numbers (which is the other part of the reals).
The fact that integers have measure 0 in the set of real numbers only seems confusing if you're thinking of everyday operations rather than thinking about mathematical abstractions.