The selection problem tries to mathematically model the following situation (Chapter III in Dynkin-Yushkevich Markov Processes): A bride-to-be is to choose among $n$ suitors who are presented one by one. She sequentially either rejects or accepts with the process stopping when she accepts one of them or when the last one is presented. Assuming she sees the first $r$ suitors, the information she obtains is their ranking. That is, a total order on $r$ elements. E.g. $r=4$ and $$ a_2< a_1 < a_4 < a_3 $$ indicates the ranking of the first $4$ suitors (the third-presented suitor being the best and the second-presented suitor being the worst). Thus, suitors lie on a linear scale (let's fix the open interval $I=(0,1)$) and ties are nonexistent.
In the text, the authors use the russian/english language to arrive at the following conclusion (page 89):
Lemma: Given that the position of the first $k$ suitors $a_1,\dots, a_k \in I$ are known, and given any component $U \subset \left(I \smallsetminus \{a_1, \dots, a_k\}\right)$ (out of the $k+1$ possibilities), we have $P[a_{k+1} \in U|a_1,\dots,a_k]=(k+1)^{-1}$.
Question: What probability space can be used to mathematically model the previous situation and the previous Lemma?
Two spaces that come to my mind are
The total space is $I^n \smallsetminus D$ with the $n$-dimensional Lebesgue measure. Here $D \subset U^n$ is the set of points $(a_1, \dots, a_n) \in I^n$ such that there exists a nontrivial permutation $\sigma$ with $$(a_{\sigma(1)}, \dots, a_{\sigma(n)}) = (a_1, \dots, a_n).$$ (We do this to prevent ties.) In this case, the Lemma does not seem to hold with $k=1$ and $a_1=1/4$. For then $a_2$ is more likely to be greater than $a_1$ than less than $a_1$.
The total space is the set of permutations $S_n$ on the first $n$ integers. Even in this case, I can't seem to square myself with the Lemma. For if $k=1$ and the permutation $a \in S_n$ has $a_1 = 1$, then the probability of $a_2$ being less than $a_1$ is zero.
I repeat my question, can someone please give me a precise probability space and an interpretation of the above Lemma within it (using integrals and conditional expectation in $\mathbb{R}^n$ if needed)? Have I misunderstood what the authors are trying to say on page 89?
Thank you for your patience. I do not understand probability.
Either of your modeling approaches will work, but the lemma is based on the idea that the only information the bride receives is the relative ranking of the suitors seen. It looks like the authors intend for the conditional probability expression $P[\dots|_1,\dots,a_k]$ to really mean $P[\dots|\sigma_k]$, where $\sigma_k$ is the relative ranking information about the first $k$ suitors. Thier choice of notation might be misleading, since the bride doesn't actually obtain the values of the $a_i$.