Why impossible events have some drawbacks or pathologies in probability theory?

Question

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure zero. The algebraic treatment gets rid of this source of unpleasantness by refusing to consider sets at all; it considers sets modulo sets of measure zero instead."

Also G. C. Rota in : Twelve problems in probability no one likes to bring up, in Algebraic Combinatorics and Computer Science : a tribute to Gian-Carlo Rota, Henry Crapo and Domenico Senato (eds.), Springer{Verlag, Milano, 2001, pp. 57{93, (1956), Math. Soc. Japan:

"Among probabilists, mention of sample points in an argument has always been bad form. A fully probabilistic argument must be pointless."

I have made some researches in order to find some consistent examples relative to the justifications of these assertions but I didn't find anything more essential.

I know that the events are not pertaining to quantum mechanics because quantum probability is a pointless probability. Morevever, we can model events by orthogonal projectors on a Hilbert Space $\mathcal{H}$, and by taking a commutative von Neumann algebra $\mathcal{M}$ together with a normal state on it as the fundamental system from which everything can be derived, as opposed to Kolmogorov's axiomatisation of classical probability theory. In the noncommutative probability theory, the pair $(\mathcal{A}, \tau)$ is termed a noncommutative probability space. It is here natural to see that kind of probability doesn't include points.

I look for concrete examples which are inadequate for probabilistic reasoning.

Answer 1

What Halmos is referring to is the development of probability theory on the basis of measure algebras. This development is spelled out in some detail by I. E. Segal in Abstract Probability Spaces and a Theorem of Kolmogoroff. Segal argues that the standard approach to probability theory introduces "tiresome mathematical complications concerning sets of measure zero" as well as other complications that are "a long way from either the practical statistical or intuitive conceptual formulation of" random variables. Intuitively, there should be no distinction between an event that is truly "impossible" and an event that occurs with probability zero, but standard measure theory does draw a distinction, which has confused generations of students, and which forces us to constantly use phrases such as "almost everywhere" and "almost surely." A development that dispenses with this largely immaterial distinction therefore enjoys a certain elegance.

It seems to me, though, that Halmos overstates the case when he claims that "all the pathology of the subject" arises from sets of measure zero. What about non-measurable sets, or random variables with undefined expected values? In any case, probabilists continue to use (and teach) standard measure theory with its sets of measure zero; if measure algebras were really such a big improvement, wouldn't probabilists have switched over en masse a long time ago? I think the reason that you're having trouble locating "decisive" examples is that there aren't any.

As for Rota's comment, I don't think it is deep. For example, if $f$ is a probability density function on the real line, then any probabilistic argument that relies essentially on the precise value of $f(x)$ at a specific point $x$ should raise a red flag, since another function $g$ that agrees with $f$ almost everywhere ought to be equivalent to $f$ as far as probability theory is concerned. But again, it may be difficult to find explicit examples of such "bad form" arguments since probabilists have trained themselves to avoid them.