## Probability: on sigma-algebra

In many probability textbooks, even at undergraduate level, statement such as “let $(\Omega, {\cal F}, \mathbb{P})$  be a probability space” is very usual. The quantity $\Omega$ is called the sample space, that is, this is the set of all possible outcomes of an “experiment”, ${\cal F}$ is a sigma-algebra (to be discussed) and finally $\mathbb{P}$  is a probability measure, a set function that assigns  a number to (specific)  subsets E of $\Omega$ that is $\mathbb{P}(E)$ is a number between 0 and 1.  A collection of subsets is called a  sigma-algebra when it satisfies the three following properties.

1. $\Omega$ is contained in ${\cal F}$.
2. ${\cal F}$  is closed by complementation, that is if E is in ${\cal F}$,  so is the complement of E.
3. ${\cal F}$ is closed by denumerable union, that is, if each $E_i$ belongs to ${\cal F}$ for i=1,2,…∞, the set F defined as the union of $E_i$  for i=1,2,…∞, that is $F=\bigcup_{i=1}^{\infty} E_i$ also belongs to ${\cal F}$.

It is then shown in many textbooks that ${\cal F}$  is also closed under denumerable intersection. To sum up, the object ${\cal F}$ is closed under complementation, denumerable intersection and denumerable unions. Most textbooks (even advanced) are unfortunately not very talkative about the further properties of a  sigma algebra ${\cal F}$ and it is even unclear to understand from the definition  whether a sigma algebra  is a “simple”  or a “complex” mathematical object. At first glance, it may seem rather simple but  it is not. Part of the difficulty  comes from the fact that one allows denumerable union and intersection rather than  finite union and intersection.  Here are rather natural questions that one can ask about a sigma algebra.

1. Is it possible to explicitly construct,  in general, a sigma algebra ?
2. Is a sigma algebra denumerable ?
3. Does  a sigma-algebra contain all the possible subsets one can think of ?

At least, the answer to these three questions is simple : No! Unless $\Omega$  a finite set, it is not possible in general to explicitly construct a sigma-algebra, see for instance the classical textbook of Bilingsley p 30-32 (click here https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf) . Consider the case in which $\Omega$ has the cardinality of the continuum, for instance $\Omega:=[0,1]$ or $\Omega:= \mathbb{R}^3$. The sigma algebra generated by the open sets of $\Omega$ is called the Borel sigma algebra and it turns out that a Borel sigma algebra  is   never denumerable (see for instance the French textbook of Revuz p 110, Mesure et intégration, Hermann, 1994). It is either a finite set (when $\Omega$  is  a finite set) or it has the cardinality of the continuum. This is far from being an intuitive result (and its proof requires to understand the theory of ordinal numbers…). The power set of $\Omega$denoted ${\cal P}(\Omega)$  is the set of all subsets of $\Omega$ and it  turns out that ${\cal F}$ is strictly contained in ${\cal P}(\Omega)$. This is once again far from being an intuitive property.

For those interested by the subject, I suggest the  lectures of Cedric Villani on Intégration et Analyse de Fourier (in French and available at http://cedricvillani.org/wp-content/uploads/2013/03/IAF.pdf), especially the topics related to the foundations of probability theory. The section entitled “Mesurabilité, non-mesurabilité, et paradoxes de Banach–Tarski” is delightful to read. On the same topic, the (French)  book of Marc Guinot entitled “Le paradoxe de Banach-Tarski” can also be very useful, and more specifically the appendix 1 entitled “Le  problème de Banach-Tarski et le problème de la mesure universelle”. Mesure universelle can be  translated as “universal measures”, and is defined as  the set function from ${\cal P}(\Omega)$ to [0,∞ ]. Guinot discusses (in appendix) various cases in which such an universal measure does not exist while Villani  directly starts page 20 with the case in which $\Omega:= \mathbb{R}^3$ and states a non-existence result, that is, an universal measure satisfying natural properties does not exist. A sigma algebra thus appears as a  “solution” for which one gets an existence result, that is, for which one can  consistenly assign a number  between zero and 1 to each  subset E of  the sigma algebra. Those subsets that belong to the sigma algebra form  the measurable subsets. For those subsets that are not measurable, it is not possible to consistently assign a probability measure. The probability measure of a non-measurable subset  can be shown to be typically  equal to zero or one, which is obviously inconsistent. Such a problem arises for instance when one considers the application of the law of large numbers with a continuum of IID random variables. See the paper by Kenneth Judd, “The law of large numbers with a continuum of IID random variables“. When $\Omega:=[0,1]$ (or $\Omega:= \mathbb{R}^3$ and more generally when $\Omega$ has the cardinality of the continuum) , ${\cal F}$ is strictly contained in ${\cal P}(\Omega)$,  which means that there are subsets in ${\cal P}(\Omega)$ that are actually not measurable.  The proof of the existence of non-measurable subsets actually makes use of the axiom of choice, an axiom at the foundation of set theory. From the discussion, the  probability theorist (that I am not)  thinks that  the statement “let $(\Omega, {\cal F}, \mathbb{P})$  be a probability space” is much more subtle than it seems, because a sigma algebra is far from being  a  simple mathematical object. For more, see the following  (difficult) book on  borel sets. https://eclass.uoa.gr/modules/document/file.php/MATH305/%CE%92%CE%BF%CE%B7%CE%B8%CE%B7%CF%84%CE%B9%CE%BA%CF%8C%20%CF%85%CE%BB%CE%B9%CE%BA%CF%8C/SynolaBorel.pdf