In introductory undergraduate probability courses, even those with a focus on set theory, I've often seen the definition of a sigma field entirely skipped over. Indeed, I've often seen the definition of a probability space include the claim "where $\mathcal{F}$ is the set of all events". Now, I've got no doubt that sigma fields are necessary in probability theory, so I'm forced to ask. How is it that these undergraduate courses can manage to teach probability theory without making any mention of sigma fields? What do they lose by skipping over this definition? Or rather, what do they have to manipulate in order to avoid it?
At the very least, I can recall from my own time as a first year undergraduate that I would occasionally get confused over what exactly an "event" is and at least in principle, I can see where "the set of all events" part of the definition could be confusing. For example, rolling a six-sided dice would give you a perfectly valid sigma field of { {6}, {not 6}, $\emptyset $, $\Omega$ } which wouldn't seem to fit the earlier definition (e.g. where's the "not 5" event?), unless you cheat the sample space in some way that I would be unsure of the validity of.
Admittedly, there's an awfully cynical part of me that thinks that the answer may be "by hoping that nobody thinks too hard about it" and indeed, inspecting my notes from such courses suggests that may genuinely be the answer. But I'm hoping that the good people at Stack Exchange can show otherwise.
Best Answer
For me, the main moral about $\sigma$-fields is that "not everything you can possibly think of is an event". That comes as a big surprise when one first encounters such situations. Fortunately, for most elementary theorems in introductory probability courses one can play the game in a rather safe way even if one has a naive idea that every subset of the probability space is measurable.
The countable additivity is, however, rather indispensable and (IMHO) should be emphasized and discussed at length, while the claim that (assuming Choice) it is incompatible with the naive idea above for plenty of natural distributions should be made but doesn't need to be discussed in depth and the curious students may be just referred to other courses and textbooks.
Poor or absent knowledge of Lebesgue integration is another major nuisance, which makes even such a natural split as $E[X]=E[X\cdot 1_{\{X<a\}}]+E[X\cdot 1_{\{X\ge a\}}]$ a technical nightmare. Here you are also often forced to say that the validity of the computations you are making will be justified elsewhere.
In short, you are completely right that it would be nice to put a rigorous foundation to everything but in practice I found that more often than not my main problems when teaching such courses are not at the level of questionable rigor but much, much lower, so the cynical advice to "just hope that nobody would notice anything" has certain merits.
As to alternative axiomatizations of probability, I'm afraid that, when done properly, they will just create a total mess in an average undergraduate student mind and make the students incapable to read standard texts afterwards, though I may be overly pessimistic here. However, the following excerpt from a review of Bruno de Finetti's textbook taken from BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 83, Number 1, January 1977 makes me wonder whether one just replaces the pre-requisite of elementary measure theory with that of advanced game theory:
After constructing a utility scale, de Finetti introduces probability via expectation, which he calls prevision. The prevision of a random variable X, "according to your opinion, is the value x which You would choose" if "You are commited to accepting any bet whatsoever with gain c(X - x) where c is arbitrary (positive or negative) at the choice of an opponent" (I, p. 87). An event E is regarded as a special case of a random variable, taking the value 0 or 1 depending on whether E is (vérifiably) false or true, and its prevision P(E) is also called its (subjective) probability, de Finetti shows that an equivalent definition of the prevision P{X) is obtained by assuming a squared loss, proportional to (X — x) , and he produces a very ingenious but elementary geometrical argument, based on squared loss, to prove the product law of probabilities (I, p. 137; and 0, p. 15).