Do probabilites not loose their effect when an infinite number of possibilities is assumed? Quantifying obervations becomes an effort in futility insofar as historic veracity is concerned. At any rate, I think I have a fairly good idea of how you come at the universe placed before you, and it is not without merit.
From my own point of view, since I am told a Creator without beginning or end is responsible for all the stuff we are able to observe, a realm of infinite possibilities is reasonable. At the same time, because the universe is organized to a degree I am able to recognize it by way of my own intelligence, I am given to believe those infinite possiblities have been purposely arranged for my viewing, albeit in weak form where my own capacites are involved.
No. Probabilities of measurements from continuous distributions have long been dealt with as limits. Most problems are solvable with Reimann's integral (the kind you learn about in 1st year calculus). There are some probability distributions it doesn't handle and the more advanced Lebesque integral from measure theory is used.