tag:blogger.com,1999:blog-3891434218564545511.post8143536260426369118..comments2024-03-28T19:56:42.305-05:00Comments on Alexander Pruss's Blog: Two spinners and infinitesimal probabilitiesAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3891434218564545511.post-62837817788085518662021-09-11T15:13:57.683-05:002021-09-11T15:13:57.683-05:00Your argument is fine but I don't see why it&#...Your argument is fine but I don't see why it's necessary: doesn't the usual proof that the measure of a point is zero suffice?<br /><br />If x is any point, and if a>0 is any number, even an infinitesimal, then x is contained in an interval of length a. If the measure/probability is uniform, then this means that<br /><br />P(x) <= a.<br /><br />Letting a->0 shows that P(x) = 0.<br /><br />But I'm not a probabilist, I'm probably missing some subtlety...Andrew Dabrowskihttps://www.blogger.com/profile/14194210589133048249noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-7890157926827176552021-09-09T10:29:20.393-05:002021-09-09T10:29:20.393-05:00Andrew: Good catch!
Ian: It's not so much the...Andrew: Good catch!<br /><br />Ian: It's not so much the naturalness of the set U that makes (3) and (4) more intuitive to me as the naturalness of the partitions.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-19871256078925787552021-09-09T09:55:57.860-05:002021-09-09T09:55:57.860-05:00"Let U consist of two line segments, one from..."Let U consist of two line segments, one from (0, 0) to (1, 1/2) and the other from (1/2, 0) to (1, 1). "<br /><br />I think you mean (0,1/2) rather than (1/2,0).Andrew Dabrowskihttps://www.blogger.com/profile/14194210589133048249noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-32203348457608511012021-09-09T01:31:25.111-05:002021-09-09T01:31:25.111-05:00Well, I can’t argue with your intuition.:-)
But ...Well, I can’t argue with your intuition.:-)<br /> <br />But in a generic independent cross product space, sloping lines are not as ‘natural’ as they could be. The natural (to me!) sets would be 'rectangles’ (i.e. a subset of A set crossed with a subset of B). The natural algebra would consist of sets that could be expressed as a finite union of such sets.<br /><br />To define general sloping straight lines (as in (2) and in the definition of U), you need special properties of A and B (viz. that you can define offset-linear functions between them.)<br /><br />So I’m not seeing (3) and (4) as more natural or intuitive than general conglomerability. If they were restricted to ‘natural’ sets as described above, maybe they would be more intuitive. But that would exclude your example U.<br /><br />All that said, conglomerability is highly intuitive in any case.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.com