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I am well aware of long history of questions in this SE regarding the difference between impossible events and events with probability of zero, former being proper subset of latter. I have read these Q&As and I am still not able to completely answer question below:

Let $r_n$ be an infinite sequence of random numbers, such that $r_n\in I=\left[0,1\right]$ drawn from uniform distribution $U_{\left[0,1\right]}$. For a given set $S\subset I$, what is the expected number of elements from $S$ to appear in $r_n$? More specifically what I'm interested in: if the set $S$ is of Lebesgue measure zero, does that imply that expected number of its elements expected in such random process is also zero?

So, if I take finite subset, expectation is obviously zero. By the fact that expectation is countably additive $E\left(\bigcup_{i}S_i\right)=\sum_i E(S_i)$ , that means that any countable subset has expectation of zero. On the other hand, if $S$ has positive measure $q$, expected number of elements appearing in $r_n$ is infinite, with $q$ as a share of sequence elements that belong to $S$. Only thing I can think of "in between" are uncountable sets of measure zero - what would be a convincing argument that those also have expectation of zero elements in such random process?

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    $\begingroup$ Regarding the impossible vs improbable, as indicated in the title: probability (mass) zero is not thought of as "impossible," as you noted, but impossible might be considered to require probability density zero as well. Your question (and the answer to it) does in fact show that in a countable sequence of observations, probability zero is "effectively impossible" in that you are not expected to ever have an observation from any probability zero set. Such a sequence of uniform reals is not expected to contain any number you can define even! (Definable reals is a countable set.) $\endgroup$
    – jdods
    Nov 1, 2022 at 18:07
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    $\begingroup$ Thanks for your comment. I was not aware of definable reals, cool concept. The more I read about real numbers, the more certain I am that the name "real" is some kind of cruel joke by mathematicians :D $\endgroup$
    – i_prob_should_know_this
    Nov 3, 2022 at 13:14

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Rather than talking about $E(S_i)$ and ranging over the elements of $S$ and asking if said element of S was seen in our sequence or not, we can range over the entries actually present in our sequence. Let $A_n$ be the indicator random variable that $r_n\in S$. Then the total number of elements of $S$ that are seen in our sequence is at most equal to $\sum\limits_{i=1}^\infty A_i$, yes? If anything, we overcounted here since there this counts duplicates multiple times. Apply linearity of expectation to this and get $0+0+0+⋯=0$ regardless of the fact that $S$ is uncountable.

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  • $\begingroup$ Thank you for your answer. Before I accept it, please suffer one more stupid question: what is cardinality of all different $A_i$? Does the last step assume countability of the set of all indicator sequences? $\endgroup$
    – i_prob_should_know_this
    Nov 3, 2022 at 13:17
  • $\begingroup$ @i_prob_should_know_this you yourself stated that $r_n$ was a sequence of random numbers. Sequences by definition contain countably many entries. For a particular sequence of random numbers, the corresponding sequence of random indicator variables will also necessarily be countable. $\endgroup$
    – JMoravitz
    Nov 3, 2022 at 13:45
  • $\begingroup$ All you said, I agree it is true. My question is: are you summing expectations not over individual elements of any particular sequence, but over set of all possible sequences? $\endgroup$
    – i_prob_should_know_this
    Nov 3, 2022 at 14:56
  • $\begingroup$ We are given a particular sequence though... the sequence $(r)_n$. $\endgroup$
    – JMoravitz
    Nov 3, 2022 at 14:58
  • $\begingroup$ Sorry, I lost you. My understanding is that only $S$ is given, not any particular $r_n$. Set of all possible outcomes of random process is $\{r_n\}$, it is equal in size to $\{A_n\}$ and its uncountable. To calculate any expectation, one should sum over that set where one cannot rely on nice properties of expectation. $\endgroup$
    – i_prob_should_know_this
    Nov 4, 2022 at 11:34

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