Probability and non-measurablity

In the measure-theoretic axioms of probability theory, probability is defined as a measure on a set of measurable subsets (called a \(\sigma\)-algebra) of the sample space. Although, in practice we (especially if we are not interested in the mathematical foundations of probability theory) do not usually care about this and intuitively expect any subset of the sample space to be measurable (i.e., to be a random event) and be capable to be assigned a non-negative value as the probability of that event to happen. In fact, this is our intuition that suggests any subset of the sample space being a random event. Until the other day when we were discussing this with Vahid, I never had thought why we do not simply replace the notion of \(\sigma\)-algebra in the definition of the probability space by the power set of the sample space. I was unconsciously convinced that we define the probability measure on a \(\sigma\)-algebra of subsets just because generally we do so in measure theory, i.e., if a collection of sets is a \(\sigma\)-algebra then of course we can define a measure on it. And moreover, I was convinced that defining the probability on a \(\sigma\)-algebra, instead of just the power set, which is by definition the biggest \(\sigma\)-algebra, is only a more formal way of defining the probability as a measure. The problem is that, in particular cases, not all of the subsets of a sample space can be measured. The Banach-Tarski theorem gives an example of such a case. According to Banach-Tarski theorem, given a \(3\)-dimensional solid ball one can decompose it into a finite number of pairwise disjoint subsets and put them back together in a different way to yield a new ball whose radius is two times bigger than the original one!

The process of reassembly involves only rotations and translations, without any stretching, bending, or adding new points. But what is the connection between this counter-intuitive result and measurability of subsets of the sample space? Let \(S\) be a \(3\)-dimensional ball whose radius is greater than \(0\) and let \(\mu(A)\) denote the volume of \(A\), where \(A\) is a subset of \(S\). Suppose we split \(S\) into disjoint subsets \(S_{1},S_{2},\dots S_{n};\, n\in\mathbb{N}\) and reassemble them, in the aforementioned manner, to get the new ball \(S^{\prime}\) with \(\mu(S^{\prime})=2\mu(S)\). The last equality implies that \[ \mu(S_{1}\cup S_{2}\cup\dots\cup S_{n})=2\mu(S_{1}\cup S_{2}\cup\dots\cup S_{n})\Rightarrow \] \[ \mu(S_{1})+\mu(S_{2})+\dots\mu(S_{n})=2[\mu(S_{1})+\mu(S_{2})+\dots\mu(S_{n})] \] Since not all of \(S_{1},S_{2},\dots,S_{n}\) are empty, one of the possible scenarios is that there exist subsets which are not measurable! As you see the possibility of existence of non-measurable sets is surely a "good" reason to restrict ourselves to \(\sigma\)-algebras for defining the probability measure, as we definitely do not want our definitions to be "formally" wrong. Apparently, doing this we are exonerated from the allegation of "being mathematically wrong". However, one can still question the validity of the Banach-Tarski theorem in the "real" world. Even more generally, does really the "mathematical world" always fit the actual world?!