I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often the sets of probability zero are in the non-discrete case sometimes a little bit contra-intuitive (like drawing a specific number from $[0,1]$ is zero, but in essence when you draw one, you get a specific number, despite its probability was zero in advance, so probability zero mean not impossible, but contrary in an intuitive interpretation in applications probability zero is almost always interpreted to mean impossible, for example when switching to an equivalent measure it is often said this measure preserved the information which events are possible. Maybe one way to think about this is a "frequentist" like interpretation, probability zero means the ratio tends to zero, so zero means not impossible [but such interpretations have often delicate "philosophical" concerns]. So what is possible is essential limited by the sample space). So my question that came to my mind:

  • the sample space essentially codes which events are impossible, for example let $\Omega = \{ H, T \}$ model the tossing of a coin, but think that the coin might land in some unusual way, like on an edge, which is extremly unlikey, but thinkable. By the $\Omega$ above this event, as not modelled, is excluded as not being possbile, but now consider $\Omega = \{ H, T, E \}$ where $E$ stands for this unrare case, and set $P(\{E\}) = 0$. So in what sense is this probability space different? With regard to this "invariant under sample space" principle both should be equivalent (and so probability theory can not distinguish between what is impossible or just extremly rare and unlikely)

  • And a more technical question: If given a probability space $(\Omega, \mathcal F, P)$, is it possible to find another probability space $(\Omega', \mathcal F', P')$ which models essentially the same (is in some sense "isomorphic") situation and such that $P$ and $P'$ are "equivalent" (i.e. $P(A) = 0$ iff $P'(A')$ where $A$ and $A'$ correspond to each other) and such that $$ P'(B) = 0 \mbox{ iff } B = \emptyset $$ in the other measure space, i.e. just the real impossible events have probability zero.

I know I am quite fuzzy in my notions (also the notion of equivalent is different from the usual one, as both $P, P'$ need not be defined over the same $\Omega$), these are just thoughts I have, maybe someone can point me to more precise notions of these ideas, if someone has already thought about them... What I have in my mind could maybe rephrased as saying, if the sample space is not that important, and probability cannot distinguish between impossible, and probability zero, then could each situtation modelled by some "cleaned up" probabibility space where the events of probability zero are indeed the impossible, i.e. the empty event?

  • $\begingroup$ What would be a "cleaned up" version of the standard Borel probability space $([0,1],\mathcal B([0,1]),\mathrm{Leb})$? $\endgroup$ – Did Jun 20 '15 at 22:24
  • $\begingroup$ I don't know, maybe someone could introduce special notions, if null sets and impossible events are somehow equivalent (up to what the theory of probability could tell) then some "equivalent" space should be possible (as here the null sets $\{x\}$ are not impossible), but maybe I am interpreting the notions wrong... $\endgroup$ – StefanH Jun 20 '15 at 22:28
  • $\begingroup$ Impossibility implies zero probability but the reverse is not true. Events of zero probability cannot be considered as equivalent. Think of the elements $x\in[0,1]$ as sets (events) for which $P(\{x\})=0$ and yet $P([0,1])>0$. $\endgroup$ – zoli Jun 21 '15 at 6:59
  • $\begingroup$ @zoli I am not sure I get the gist of your comment. Actually I read three rather unrelated sentences, the first one and the second one that I fail to understand, and the last one that I understand but which seems unrelated to the question. Surely I am too slow... $\endgroup$ – Did Jun 21 '15 at 10:43
  • $\begingroup$ @Did: Sentence No. 1 carefully formulated: "If an even is an impossible event (a logically impossible event) then its probability is necessarily zero." Sentence No. 2: "The reverse statement is not true: If the probability of an event is zero then the event is not necessarily impossible." Indeed, sentence No. 3. is not related to the first two sentences. It is an example trying to show that events of zero probability cannot be considered as some kind of equivalence class, as the OP suggests; their topological union does not look like an impossible event. $\endgroup$ – zoli Jun 21 '15 at 16:41

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