Possibility ≠ Probability … but how big is the difference?

Geometry Level 5

Consider a $1m\times 1m\times 1m$ region $\Omega$ of space in the universe. Consider the random event of a particle appearing in the region $\Omega$ . (Assume that particles are idealised and occupy perfect points in space, when they appear.) Let $\mathbb{P}[\cdot]$ denote a probability distribution of where the particle appears.

Observation 1. If we take for example the uniform distribution, the probability measure has the continuity property that $\mathbb{P}[\{x\}]=0$ for each $x\in\Omega$ . That is, the event $\{x\}$ is improbable . On the other hand the particle can appear at the point $x$ —it is possible . This illustrates that there is distinction between possibility (a metaphyisical concept) and probability (an empirical concept). One never encounters these distinctions in finite/discrete spaces, and we shall explore this further in this problem. Above we discussed small possibilities. We now turn our attention to events that represent large possibilities.

Definition 1. A set $E\subseteq\Omega$ is called comeagre iff it contains a countable intersection of dense open subsets, that is $E\supseteq\bigcap_{n\in\mathbb{N}}U_{n}$ for some dense, open subsets $U_{n}\subseteq\Omega$ .

Note, that for spaces like $\Omega$ , so-called Baire-spaces, comeagre subsets are huge . They are dense, and one can intersect countable infinitely many of them, and they still remain comeagre.

Definition 2. Call an event $E$

impossible iff $E=\emptyset$
improbable iff $\mathbb{P}[E]=0$
possible iff $E\neq\emptyset$
probable iff $\mathbb{P}[E]>0$
almost certain iff $\mathbb{P}[E]=1$
topologically certain iff $E$ is comeagre
certain iff $E=\Omega$ .

Observation 2. Clearly we have

impossible $\Longrightarrow$ improbable ;
certain $\Longrightarrow$ almost certain ;
certain $\Longrightarrow$ topologically certain ;
probable $\Longrightarrow$ possible .

We also have by the above discussion, that the reverse implications are not so clear cut. For example, improbable $\not\Longrightarrow$ impossible . QUESTION. But what about topologically certain events? It would be nice to know if they are almost certain . But I will ask a simpler question: are topologically certain events probable ? Draw your metaphyisical/philosophical conclusions in the comments below!

NB: Assume you don’t know what the probability distribution of the appearance of the particle in the region of space is. Assume that you only know, that it has the continuity-property (see above). Further properties of (probability) measures are:

$\sigma$ -(sub)additivity: $\mathbb{P}[\bigcup_{n\in\mathbb{N}}A_{n}]\leq\sum_{n\in\mathbb{N}}\mathbb{P}[A_{n}]$ for any measurable events $A_{0},A_{1},\ldots\subseteq\Omega$ , with equality if the events are mutually exclusive;
monotonicity: $\mathbb{P}[E]\leq\mathbb{P}[E']$ for any measureable events $E\subseteq E'\subseteq\Omega$ ;
continuity-from-above: $\mathbb{P}[\bigcap_{n\in\mathbb{N}}E_{n}]=\inf_{n}\mathbb{P}[E_{n}]$ for any measurable events $\Omega\supseteq E_{0}\supseteq E_{1}\supseteq \ldots$ ;
probabilistic: $\mathbb{P}[\Omega]=1$ ;
continuity: $\mathbb{P}[\{x\}]=0$ for all $x\in\Omega$ .

Note that some of these properties are reducible to others.

For any distribution: never. Depending on the distribution: not always. For any distribution: always. For any distribution: not always. Depending on the distribution: never. Depending on the distribution: always.

1 solution

R Mathe
May 27, 2018

The answer is: for any distribution not always.

Fix any distribution $\mathbb{P}$ with the continuity-property.

Part I. Some topologically certain events are probable : To prove this, consider the certain event, $E=\Omega$ . This is both topologically certain and almost certain . From the latter it follows that it is probable .

Part II. Some topologically certain events are improbable : At first sight, one might think, How can this be? A massive set, which has points everywhere, ought to have probability $1$ or at least have probability $>0$ . Well, this is not the case. In general, it is a fallacy to think, that ubiquity in one sense implies ubiquity or even significance in another. We shall construct an event, $E$ , which is topologically certain but for which $\mathbb{P}[E]=0$ .

Surprisingly (or maybe intuitively?) this construction does not even depend on the underlying space, except that it be metrisable and have countable dense subset. That is why I fixed for simplicity’s sake a space like $\Omega=[0,1]\times[0,1]\times[0,1]$ . Anyhow, so long as the space satisfies the afore mentioned (as does $\Omega$ ), one may

fix a compatible metric $d$ and a countable dense subset $D:=\{x_{n}:n\in\mathbb{N}\}\subseteq\Omega$ .

We will

construct sets $U_{n}=\bigcup_{k\in\mathbb{N}}\mathcal{B}(x_{k};r_{n,k})$ with $r_{n,k}\in\mathbb{R}^{+}$

which are open and dense (since they contain the dense subset $D$ ). Then we

set $E:=\bigcap_{n\in\mathbb{N}}U_{n}$

which is then by definition comeagre, hence topologically certain . It remains to choose the radii in such a way, that $\mathbb{P}[E]=0$ .

Side calculation. Fix $x\in\Omega$ . The properties of a metric space yield that $\bigcap_{\delta\in\mathbb{Q}^{+}}\mathcal{B}(x;\delta)=\{x\}$ . By the continuity requirement $0=\mathbb{P}[\{x\}]=\mathbb{P}[\bigcap_{\delta\in\mathbb{Q}^{+}}\mathcal{B}(x;\delta)]$ . On the other hand $\mathbb{P}[\bigcap_{\delta\in\mathbb{Q}^{+}}\mathcal{B}(x;\delta)]=\inf_{\delta\in\mathbb{Q}^{+}}\mathbb{P}[\mathcal{B}(x;\delta)]$ . Thus $\inf_{\delta\in\mathbb{Q}^{+}}\mathbb{P}[\mathcal{B}(x;\delta)]=0$ . It follows that for all $\varepsilon>0$ there exists an $\delta_{x,\varepsilon}>0$ , such that $\mathbb{P}[\mathcal{B}(x;\delta_{x,\varepsilon})]<\varepsilon$ .

Construction.

Set $r_{n,k}:=\delta_{x_{k},2^{-(n+k+1)}}$ for all $n,k\in\mathbb{N}$ .

Then one has

$\mathbb{P}[U_{n}] \leq \sum_{k\in\mathbb{N}}\mathbb{P}[\mathcal{B}(x;r_{n,k})] \leq \sum_{k\in\mathbb{N}}2^{-(n+k+1)}=2^{-n}$

and thus

$\mathbb{P}[E]=\mathbb{P}[\bigcap_{n\in\mathbb{N}}U_{n}] \leq\inf_{n\in\mathbb{N}}\mathbb{P}[U_{n}] \leq\inf_{n\in\mathbb{N}} 2^{-n}=0$

Hence $\mathbb{P}[E]=0$ . Thus $E$ is an topologically certain event, which occurs with probability $0$ , ie it is improbable. This completes the proof, that some topologically certain events are improbable .

Remark. My philosophical conclusion is, that not only is there a discrepancy between being possible and being probable, this gap is huge: there are events which through the lens of topology comprise certainties, but which empirically constitute improbable events. This is troubling, as it is not clear, through what lens one should view the world. Alternatively this means that there are events which take place everywhere, yet which go completely unnoticed by certain observers. This has further implications towards the realm of perception: how do we know exactly through what filter we observe reality? Geometrically (highlighting topologically certain phenomena) or empirically (highlighting probabilistically certain phenomena)? Or does it suggest, that these infinite structures cannot / do not manifest themselves in nature?

Hi! Is this question original? Is there any book you'd recommend about this subject?

Lucas Viana Reis - 2 years, 4 months ago

Yes and no. The focus, question and observations are completely from me. That there are comeagre sets of probability 0 is known. But the focus isn't really on this fact. In literature on descriptive set theory , the more general theorem-like similarities and structural disparities in measure vs. category is well studied. So, you can find this in books covering descriptive set theory and measure theory in chapters comparing category and measure.

R Mathe - 2 years, 4 months ago

Possibility ≠ Probability … but how big is the difference?

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