tag:blogger.com,1999:blog-3891434218564545511.post3393681626145275200..comments2024-03-28T19:56:42.305-05:00Comments on Alexander Pruss's Blog: Domination and uniform spinnersAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-3891434218564545511.post-50051139630632199622022-02-18T20:09:29.062-06:002022-02-18T20:09:29.062-06:00The construction in this post is pretty much the s...The construction in this post is pretty much the same as John Norton's Vitali set lottery: https://sites.pitt.edu/~jdnorton/papers/Infinite_lottery_not_final.pdfAlexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-55843370167780620592022-02-18T11:42:42.797-06:002022-02-18T11:42:42.797-06:00BTW, if you take a model of set theory with counta...BTW, if you take a model of set theory with countable choice but where all sets of reals are Lebesgue measurable, then there will be a partial preference ordering on wagers on the outcomes of a spinner that (a) is strongly rotation invariant (W1 is at least as good as W2 iff rW1 is at least as good as W2 iff W1 is at least as good as rW2) and (b) satisfies condition (2) (but not condition (1)): wager W1 is at least as good as W2 iff the Lebesgue integral of W1-W2 is defined and non-negative. For every positive function will have a well-defined Lebesgue integral. <br /><br />I don't know if there is a model of set theory where there is a total preference ordering on wagers on the outcomes of a spinner that satisfies (a) and (b). Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-60942034987893423262022-02-18T11:25:50.807-06:002022-02-18T11:25:50.807-06:00"The post could be taken as ‘constructing’ a ..."The post could be taken as ‘constructing’ a fair infinite lottery." I didn't notice that. Cool!Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-23041479652830335552022-02-18T03:14:56.714-06:002022-02-18T03:14:56.714-06:00By one of your theorems (Non-classical Probabiliti...By one of your theorems (Non-classical Probabilities Invariant Under Symmetries, Th 2), there is no rotation-invariant regular hyperreal distribution on the circle. So it’s not so clear how a ‘spinner’ should be represented mathematically. You have to accept that if you force rotation-invariance on some sort of sets, you can’t have it on others. The example illustrates this.<br /><br />Another way of looking at it: If X is the outcome of a fair infinite lottery on the integers, then X+1 is also fair, and is always greater than X. (<i>Fair</i> here means <i>fair between singletons</i>. This does not imply <i>translation invariant for arbitrary sets</i>.) The post could be taken as ‘constructing’ a fair infinite lottery.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-70798554415864498922022-02-17T20:03:08.854-06:002022-02-17T20:03:08.854-06:00Let rz be z rotated by x degrees. Then let g(z) = ...Let rz be z rotated by x degrees. Then let g(z) = u(rz)-u(z) be the amount gained by the rotation.<br /><br />In the main construction, where u(z) is unbounded, g(z) = 1 for every z. Constant gain everywhere. But if we make u bounded (say by the arctan move), then interestingly I think I can prove that although g(z) is strictly positive, it is (a) non-measurable, and (b) there is no strictly positive measurable function less than g(z). This makes the bounded case, to me, a bit less impressive.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-40616027141065609092022-02-17T13:40:58.436-06:002022-02-17T13:40:58.436-06:00One can also do a similar, but not quite the same,...One can also do a similar, but not quite the same, thing with infinite sequences of coin flips. In other words, there is (assuming AC) a game based on a bidirectionally infinite sequence of fair coin flips (arranged left-to-right) with the property that there is an event E such that:<br /> 1. P(E) = 0 (indeed, E is countable)<br /> 2. the game obtained by shifting the coins one space to the right always pays at least as well<br /> 3. the game obtained by shifting the coins one space to the right pays better outside of E.<br /><br />Here, E is the event that the coin flip outcomes are an endlessly repeating pattern (e.g., ...HHTHHTHHT...). We prove the above by letting ~ be the equivalence relation that holds between two sequences of coin flips when one sequence is a shift of the other, and then running an argument much as in the post, though it only works outside E.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.com