{"id":1597,"date":"2018-11-26T19:35:15","date_gmt":"2018-11-26T18:35:15","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1597"},"modified":"2022-11-19T15:15:16","modified_gmt":"2022-11-19T14:15:16","slug":"the-locale-of-random-elements-of-a-space","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1597","title":{"rendered":"The locale of random elements of a space"},"content":{"rendered":"<p><a href=\"https:\/\/homepages.inf.ed.ac.uk\/als\/\">Alex Simpson<\/a> has a lot of slides with very interesting ideas. \u00a0One of them is what he calls the <a href=\"https:\/\/homepages.inf.ed.ac.uk\/als\/Talks\/ccc09.pdf\">locale of random sequences<\/a>. \u00a0(Update, November 19th, 2022: this link now points nowhere. \u00a0I am unable to find his slides nowadays. \u00a0However, look at [1].) \u00a0This is a terribly clever idea that aims at solving the question &#8220;what are random sequences?&#8221;, using locale theory.<\/p>\n<p>I will not repeat the first 18 slides, which explain the problem: have a look by <a href=\"https:\/\/homepages.inf.ed.ac.uk\/als\/Talks\/ccc09.pdf\">yourselves<\/a>.<\/p>\n<p>However, simply imagine you are given an infinite sequence, or a real number between 0 and 1. \u00a0Is it random? \u00a0There are notions like Kolmogorov randomness which will tell you whether it is or not, in a very precise sense, but the simplest answer is <em>no<\/em>: since you know what it is, and you can always read the next bit or the next digit, it is completely deterministic, hence not random. \u00a0Alex&#8217;s brilliant idea is that the space of random sequences (or numbers)\u00a0<em>cannot have points<\/em>. \u00a0And a nice way to produce a non-trivial space without points is as a locale.<\/p>\n<p>The construction itself is pretty simple. \u00a0Consider a topological space\u00a0<em>X<\/em>, which may be Cantor space (for random sequences) or [0, 1] (for random numbers) for example, and let \u03bc be some continuous probability valuation on\u00a0<em>X<\/em> (I will give the definition below, but imagine a uniform distribution). \u00a0Define an equivalence relation on the frame\u00a0<strong>O<\/strong><em>X<\/em> of open subsets of\u00a0<em>X<\/em> by\u00a0<em>U<\/em>\u2261<em>V<\/em> if and only if \u03bc(<em>U<\/em>\u222a<em>V<\/em>)=\u03bc(<em>U<\/em>\u2229<em>V<\/em>), and then build the sublocale\u00a0<strong>O<\/strong><em>X<\/em>\/\u2261 (see below). \u00a0I will try to explain why this is a good idea below, but first, a few definitions.<\/p>\n<h2>Continuous valuations<\/h2>\n<p>A continuous valuation \u03bc is something like a measure, but better suited to topological spaces: it is a map from\u00a0<strong>O<\/strong><em>X<\/em> to\u00a0\u211d<sub>+<\/sub>\u00a0\u222a {\u221e} that is:<\/p>\n<ul>\n<li>strict (\u03bc(\u2205)=0),<\/li>\n<li>monotonic (if\u00a0<em>U<\/em> is included in\u00a0<em>V<\/em>, then \u03bc(<em>U<\/em>)\u2264\u03bc(<em>V<\/em>)),<\/li>\n<li>modular (\u03bc(<em>U<\/em>)+\u03bc(<em>V<\/em>)=\u03bc(<em>U<\/em>\u222a<em>V<\/em>)+\u03bc(<em>U<\/em>\u2229<em>V<\/em>)),<\/li>\n<li>and Scott-continuous (for every directed family of open sets\u00a0<em>U<sub>i<\/sub><\/em>, <em>i<\/em>\u2208<em>I<\/em>, \u03bc(\u222a<em><sub>i<\/sub><\/em><em>U<sub>i<\/sub><\/em>)=sup<em><sub>i<\/sub><\/em> \u03bc(\u222a<em>U<sub>i<\/sub><\/em>)).<\/li>\n<\/ul>\n<p>I will sometimes say that <em>U<\/em> has\u00a0<em>\u03bc-measure<\/em> equal to\u00a0<em>a<\/em> if\u00a0\u03bc(<em>U<\/em>)=<em>a<\/em>. \u00a0If you are afraid that this does not quite look like a measure, please note that (\u03c3-finite) measures and (\u03c3-finite) continuous valuations are in one-to-one correspondence on Polish spaces, and even more, on all completely metrizable spaces. \u00a0This and much more can be found in [3] (well, what I have just said is there, but is pretty much hidden: read Corollary 4.5 there, together with the first paragraph of Section 4 for the existence of measure extensions in the case of completely metrizable spaces; read the final paragraph of that same section for uniqueness).<\/p>\n<p>A continuous\u00a0<em>probability valuation<\/em> additionally satisfies\u00a0\u03bc(<em>X<\/em>)=1. \u00a0I will not require that, but I will need\u00a0\u03bc to be\u00a0<em>bounded<\/em>, namely to satisfy\u00a0\u03bc(<em>X<\/em>)&lt;\u221e. \u00a0 It is likely that one can push what will follow to locally finite continuous valuations, namely to continuous valuations such that every point has an open neighborhood of finite\u00a0\u03bc-measure, but I will not pursue this.<\/p>\n<p>Lebesgue measure on [0, 1] defines a continuous probability valuation, when restricted to the open sets, for example. \u00a0In general, every\u00a0<em>\u03c4-smooth<\/em>\u00a0probability measure does\u2014a\u00a0\u03c4-smooth measure is, by definition, a measure that is Scott-continuous once restricted to the open sets.<\/p>\n<h2>The need for a sublocale<\/h2>\n<p>The sublocale\u00a0<strong>O<\/strong><em>X<\/em>\/\u2261 is certainly not the set of equivalence classes of open subsets of\u00a0<em>X<\/em> modulo\u00a0\u2261. \u00a0That would be the order-theoretic quotient. \u00a0What we really want to build is something like a <em>subspace<\/em> of\u00a0<em>X<\/em>, not\u00a0<strong>O<\/strong><em>X<\/em>, inside the category of locales. \u00a0Hence we should build a sublocale\u2014or, preferably, a <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=940\">nucleus<\/a>, since that will be easier\u2014, not a quotient. \u00a0One way of doing so is by defining a suitable equivalence relation\u00a0\u2261 on <strong>O<\/strong><em>X<\/em>, and defining the corresponding nucleus as mapping every open set\u00a0<em>U<\/em> to the largest open set equivalent to\u00a0<em>U<\/em>, then the desired sublocale as the set of fixed points of that nucleus.<\/p>\n<p>As I have already mentioned, we will define \u2261 by:\u00a0<em>U<\/em>\u2261<em>V<\/em> if and only if \u03bc(<em>U<\/em>\u222a<em>V<\/em>)=\u03bc(<em>U<\/em>\u2229<em>V<\/em>). \u00a0The idea is that we would like to build the &#8220;subspace of random elements&#8221; as the &#8220;subspace&#8221; of those points\u00a0<em>x<\/em> such that for every open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em>, every smaller open neighborhood\u00a0<em>V<\/em>\u00a0with the same\u00a0\u00a0\u03bc-measure\u00a0should also contain\u00a0<em>x<\/em>. \u00a0This is equivalent to saying that if\u00a0<em>x<\/em> is in\u00a0<em>U<\/em>, then\u00a0<em>x<\/em> is in every open set equivalent to\u00a0<em>U<\/em>.<\/p>\n<p>As we have already announced, the resulting &#8220;subspace&#8221; will not have any point, and this justifies the quotes. \u00a0In fact, if we do not use locales, there is no non-empty subspace satisfying the above property in general. \u00a0For example, with the Lebesgue measure on [0, 1], for any &#8220;random element&#8221;\u00a0<em>x<\/em>, take\u00a0<em>U<\/em>=[0, 1] and\u00a0<em>V<\/em>=[0,\u00a0<em>x<\/em>)\u00a0\u222a (x, 1]:\u00a0<em>U<\/em> and\u00a0<em>V<\/em> have the same measure but one contains\u00a0<em>x<\/em> and the other does not.<\/p>\n<h2>The sublocale of random elements as a nucleus<\/h2>\n<p>Recall that <em>U<\/em>\u2261<em>V<\/em> if and only if \u03bc(<em>U<\/em>\u222a<em>V<\/em>)=\u03bc(<em>U<\/em>\u2229<em>V<\/em>).\u00a0 We first check that:<\/p>\n<p><strong>Lemma 1.<\/strong>\u00a0\u2261 is an equivalence relation on <strong>O<\/strong><em>X<\/em>.<\/p>\n<p>Proof. The only non-trivial fact to check is transitivity. \u00a0Let us assume\u00a0<em>U<\/em>\u2261<em>V<\/em> and\u00a0<em>V\u2261W<\/em>. \u00a0Since\u00a0<em>U<\/em>\u2261<em>V<\/em>,\u00a0<em>U<\/em>\u222a<em>V<\/em>\u00a0and\u00a0<em>U<\/em>\u2229<em>V<\/em>\u00a0have the same\u00a0\u03bc-measure. \u00a0By monotonicity, the\u00a0\u03bc-measures of\u00a0<em>U<\/em> and of\u00a0<em>V<\/em> are inbetween. \u00a0Therefore\u00a0<em>U<\/em>\u222a<em>V<\/em>,\u00a0<em>U<\/em>\u2229<em>V<\/em>,\u00a0<em>U<\/em>, <em>V<\/em> all have the same\u00a0\u03bc-measure. \u00a0Similarly, <em>V<\/em>\u222a<em>W<\/em>,\u00a0<em>V<\/em>\u2229<em>W<\/em>,\u00a0<em>V<\/em>, <em>W<\/em>\u00a0all have the same\u00a0\u03bc-measure. \u00a0Since\u00a0<em>V<\/em> is in both lists, all those sets have the same\u00a0\u03bc-measure. \u00a0Call it <em>a<\/em>. \u00a0Now:<\/p>\n<p>2<em>a<\/em> = \u03bc(<em>U<\/em>\u2229<em>V<\/em>)+\u03bc(<em>V<\/em>\u2229<em>W<\/em>)<br \/>\n=\u00a0\u00a0\u03bc((<em>U<\/em>\u2229<em>V<\/em>)\u222a(<em>V<\/em>\u2229<em>W<\/em>))+\u03bc((<em>U<\/em>\u2229<em>V<\/em>)\u2229(<em>V<\/em>\u2229<em>W<\/em>)) [modularity]<br \/>\n=\u00a0\u00a0\u03bc(<em>V<\/em>\u2229(<em>U<\/em>\u222a<em>W<\/em>))+\u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2229<em>W<\/em>)<br \/>\n\u2264 2<em>a<\/em> [monotonicity],<br \/>\nso all those values are equal. \u00a0In particular,\u00a0\u03bc(<em>V<\/em>\u2229(<em>U<\/em>\u222a<em>W<\/em>))+\u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2229<em>W<\/em>)=2<em>a<\/em>. \u00a0By monotonicity, \u03bc(<em>V<\/em>\u2229(<em>U<\/em>\u222a<em>W<\/em>)) and \u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2229<em>W<\/em>) are both less than or equal to <em>a<\/em>. \u00a0Since\u00a0\u03bc is bounded, <em>a<\/em>\u2260\u221e, so they are both equal to\u00a0<em>a<\/em>. \u00a0Then all the open sets inbetween\u00a0<em>U<\/em>\u2229<em>V<\/em>\u2229<em>W<\/em> and\u00a0<em>U<\/em> (or\u00a0<em>W<\/em>) have the\u00a0same\u00a0\u03bc-measure, in particular <em>U<\/em>\u2229<em>W<\/em>. \u00a0Finally,\u00a02<em>a<\/em> = \u03bc(<em>U<\/em>)+\u03bc(<em>W<\/em>) =\u00a0\u03bc(<em>U<\/em>\u2229<em>W<\/em>)+\u03bc(<em>U<\/em>\u222a<em>W<\/em>) =\u00a0<em>a<\/em>+\u03bc(<em>U<\/em>\u222a<em>W<\/em>) by modularity, so\u00a0\u03bc(<em>U<\/em>\u222a<em>W<\/em>) is also equal to\u00a0<em>a<\/em>, hence to \u03bc(<em>U<\/em>\u2229<em>W<\/em>). \u00a0\u2610<\/p>\n<p><strong>Lemma 2.<\/strong> If <em>U<\/em>\u2261<em>V<\/em>\u00a0and\u00a0<em>U<\/em>\u2261<em>V&#8217;<\/em>\u00a0then\u00a0<em>U<\/em>\u2261<em>V<\/em>\u222a<em>V&#8217;<\/em>\u00a0and\u00a0<em>U<\/em>\u2261<em>V<\/em>\u2229<em>V&#8217;<\/em>.<\/p>\n<p>Proof. \u00a0By transitivity (Lemma 1), <em>V<\/em>\u2261<em>V&#8217;<\/em>, so\u00a0<em>V<\/em>\u222a<em>V&#8217;<\/em>,\u00a0<em>V<\/em>\u2229<em>V&#8217;<\/em>,\u00a0<em>V<\/em>, <em>V&#8217;<\/em>\u00a0all have the same\u00a0\u03bc-measure. \u00a0Note that for any two open subsets <em>A<\/em> and\u00a0<em>B<\/em> such that\u00a0<em>A<\/em> is included in\u00a0<em>B<\/em>,\u00a0<em>A<\/em>\u2261<em>B<\/em> if and only if\u00a0\u03bc(<em>A<\/em>)=\u03bc(<em>B<\/em>). \u00a0Hence\u00a0<em>V<\/em>\u222a<em>V&#8217;<\/em>\u2261<em>V<\/em>,\u00a0<em>V\u222aV&#8217;<\/em>\u2261<em>V&#8217;<\/em>, <em>V<\/em>\u2261<em>V<\/em>\u2229<em>V&#8217;<\/em>, and\u00a0<em>V&#8217;<\/em>\u2261<em>V<\/em>\u2229<em>V&#8217;<\/em>. \u00a0We conclude by Lemma 1. \u00a0\u2610<\/p>\n<p><strong>Proposition 3.<\/strong>\u00a0 For every open subset\u00a0<em>U<\/em> of\u00a0<em>X<\/em>, there is a largest open subset\u00a0\u03bd(<em>U<\/em>) of\u00a0<em>X<\/em>\u00a0such that\u00a0<em>U<\/em>\u2261\u03bd(<em>U<\/em>). \u00a0The map \u03bd :\u00a0<em>U<\/em>\u00a0\u21a6\u00a0\u03bd(<em>U<\/em>) is a nucleus.<\/p>\n<p>Proof. \u00a0Let\u00a0<em>D<\/em> be the family of open subsets\u00a0<em>V<\/em> of\u00a0<em>X<\/em> such that\u00a0<em>U<\/em>\u2261<em>V<\/em>. \u00a0It is non-empty, since it contains\u00a0<em>U<\/em>. \u00a0Given two elements\u00a0<em>V<\/em> and\u00a0<em>V&#8217;<\/em> of\u00a0<em>D<\/em>,\u00a0<em>V<\/em>\u222a<em>V&#8217;<\/em> is also in\u00a0<em>D<\/em> by Lemma 2, first part. \u00a0This shows that\u00a0<em>D<\/em> is directed. \u00a0Note that all the elements\u00a0<em>V<\/em> of\u00a0<em>D<\/em> have the same\u00a0\u03bc-measure, which is also equal to\u00a0\u03bc(<em>U<\/em>)<em>. \u00a0<\/em>Since\u00a0<em>D<\/em> is directed, it has a supremum, which is the union of the elements of\u00a0<em>D<\/em>. \u00a0We call it \u03bd(<em>U<\/em>). \u00a0Since \u03bc is Scott-continuous, we obtain that\u00a0\u03bc(\u03bd(<em>U<\/em>)) is the supremum of the \u03bc-measures of elements of\u00a0<em>D<\/em>, which are all equal to\u00a0\u03bc(<em>U<\/em>). \u00a0Since\u00a0<em>U<\/em> is included in \u03bd(<em>U<\/em>),\u00a0\u03bc(<em>U<\/em>\u222a\u03bd(<em>U<\/em>))=\u03bc(\u03bd(<em>U<\/em>))=\u03bc(<em>U<\/em>)=\u03bc(<em>U<\/em>\u2229\u03bd(<em>U<\/em>)), so\u00a0<em>U<\/em>\u2261\u03bd(<em>U<\/em>). \u00a0It follows that\u00a0\u03bd(<em>U<\/em>) is in\u00a0<em>D<\/em>, hence is the largest element of\u00a0<em>D<\/em>.<\/p>\n<p>It is easy to see that\u00a0\u03bd :\u00a0<em>U<\/em>\u00a0\u21a6\u00a0\u03bd(<em>U<\/em>) is monotonic, inflationary, and idempotent. It remains to show that\u00a0\u03bd(<em>U<\/em>\u2229<em>V<\/em>)=\u03bd(<em>U<\/em>)\u2229\u03bd(<em>V<\/em>) for all open sets\u00a0<em>U<\/em> and\u00a0<em>V<\/em>. \u00a0To that end, it is enough to show that\u00a0\u03bd(<em>U<\/em>\u2229<em>V<\/em>) contains \u03bd(<em>U<\/em>)\u2229\u03bd(<em>V<\/em>), since the other direction follows from monotonicity. \u00a0That amounts to showing that every open set\u00a0<em>W<\/em>\u00a0such that\u00a0<em>W<\/em>\u2261<em>U<\/em> and\u00a0<em>W<\/em>\u2261<em>V<\/em> also satisfies\u00a0<em>W<\/em>\u2261<em>U<\/em>\u2229<em>V<\/em>: this is given by Lemma 2, second part. \u00a0\u2610<\/p>\n<h2>The sublocale of random elements as a sieve<\/h2>\n<p>Given a nucleus \u03bd on a frame \u03a9, we recall that it can be represented in an alternate way as a <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=908\">sieve<\/a>\u00a0<em>S<sub>\u03bd<\/sub><\/em>, defined as the set of formal crescents (<em>u<\/em>, <em>v<\/em>) such that <em>u<\/em> \u2270 \u03bd(<em>v<\/em>). \u00a0Here\u00a0\u03a9=<strong>O<\/strong><em>X<\/em>, and\u00a0<em>u<\/em> and\u00a0<em>v<\/em> are actual open subsets\u00a0<em>U<\/em> and <em>V.<\/em><\/p>\n<p><strong>Lemma 4.<\/strong>\u00a0<em>U<\/em>\u00a0\u228a\u00a0\u03bd(<em>V<\/em>) if and only if \u03bc(<em>U<\/em>)&gt;\u03bc(<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>).<\/p>\n<p>Proof. \u00a0<em>U<\/em> \u2286 \u03bd(<em>V<\/em>) iff\u00a0\u03bd(<em>U<\/em>) \u2286 \u03bd(<em>V<\/em>) [using the fact that \u03bd is a nucleus] iff\u00a0\u03bd(<em>U<\/em>) = \u03bd(<em>U<\/em>) \u2229\u00a0\u03bd(<em>V<\/em>) iff\u00a0\u03bd(<em>U<\/em>) = \u03bd(<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>) iff <em>U<\/em>\u2261<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>\u00a0iff\u00a0\u03bc(<em>U<\/em>)=\u03bc(<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>). \u00a0\u2610<\/p>\n<p>There is a theorem due independently to Pettis, Smiley, and Horn-Tarski which says that every bounded valuation \u03bc (not necessarily continuous) extends to an additive measure, again written \u03bc, on the smallest Boolean algebra of subsets containing the open sets. \u00a0The elements of that algebra are the finite disjoint unions of crescents <em>U<\/em>\u2013<em>V<\/em>. \u00a0What Lemma 4 says is that <em>S<sub>\u03bd<\/sub><\/em>\u00a0is the set of formal crescents (<em>U<\/em>, <em>V<\/em>) such that\u00a0\u03bc(<em>U<\/em>\u2013<em>V<\/em>)\u22600.<\/p>\n<p>I will not formally need that theorem, but I will reuse the convention of writing\u00a0\u03bc(<em>U<\/em>\u2013<em>V<\/em>)\u22600 instead of\u00a0\u03bc(<em>U<\/em>)&gt;\u03bc(<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>). \u00a0That is somehow more readable.<\/p>\n<p>Since a sieve is a collection of formal crescents that is closed under equivalence of formal crescents, and equivalence of formal crescents of the form (<em>U<\/em>, <em>V<\/em>)\u00a0is just equality of the corresponding crescents <em>U<\/em>\u2013<em>V<\/em>, an alternative description of <em>S<sub>\u03bd<\/sub><\/em>\u00a0is as the collection of (actual) crescents <em>S<sub>\u03bd<\/sub><\/em>\u00a0of non-zero\u00a0\u03bc-measure.<\/p>\n<h2>The points of the sublocale of random elements<\/h2>\n<p>We have already argued that a sub<em>space<\/em> of random elements would have to be empty, at least on [0, 1] with Lebesgue measure. \u00a0Correspondingly, the sublocale of random elements has no points in general, as we shall see.<\/p>\n<p>First, any nucleus \u03bd determines a sublocale \u03a9<em><sub>\u03bd<\/sub><\/em>, consisting of the fixed points (equivalently, the image) of the nucleus. \u00a0Every sublocale is a frame (exercise, or see [2, Proposition 2.2]), hence it makes sense to explore what its points are.<\/p>\n<p>We work directly with the nucleus \u03bd. \u00a0A nucleus preserves all finite infima, so the finite infima of \u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0are computed as in the ambient frame\u2014in our case, as finite intersections. \u00a0Suprema, on the other hand, are computed by taking suprema in the ambient frame (union, in our case) and then taking the image by \u03bd.<\/p>\n<p>A point of \u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0is described by a prime element of \u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0(Proposition 8.1.20 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>). \u00a0Here this is an open subset\u00a0<em>P<\/em> of\u00a0<em>X<\/em> that is different from\u00a0<em>X<\/em>, inside \u00a0\u03a9<em><sub>\u03bd\u00a0<\/sub><\/em>(namely, <i>P<\/i>=\u03bd(<em>P<\/em>)), and such that for all open subsets\u00a0<em>U<\/em>,\u00a0<em>V<\/em> in \u03a9<em><sub>\u03bd\u00a0<\/sub><\/em>(i.e., such that\u00a0<em>U<\/em>=\u03bd(<em>U<\/em>) and <em>V<\/em>=\u03bd(<em>V<\/em>)), if\u00a0<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>\u00a0\u2286 <em>P<\/em> then\u00a0<em>U<\/em> or\u00a0<em>V<\/em> is included in\u00a0<em>P<\/em>.<\/p>\n<p><strong>Proposition 5.<\/strong> The prime elements of\u00a0\u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0are the open subsets of the form\u00a0\u03bd(<em>P<\/em>), where\u00a0<em>P<\/em> is an open subset of\u00a0<em>X<\/em> such that for all open subsets\u00a0<em>U<\/em> and\u00a0<em>V<\/em> of\u00a0<em>X<\/em> such that\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 and\u00a0\u03bc(<em>V\u2013<\/em><em>P<\/em>)\u22600, then\u00a0\u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2013<em>P<\/em>)\u22600.<\/p>\n<p>Proof. Assume\u00a0<em>P<\/em> prime in \u03a9<em><sub>\u03bd<\/sub><\/em>,\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 and\u00a0\u03bc(<em>V\u2013<\/em><em>P<\/em>)\u22600. \u00a0By Lemma 4, neither\u00a0<em>U<\/em> nor\u00a0<em>V<\/em> is included in\u00a0\u03bd(<em>P<\/em>). \u00a0Then neither \u03bd(<em>U<\/em>) nor \u03bd(<em>V<\/em>) is included in \u03bd(<em>P<\/em>). \u00a0Since \u03bd(<em>U<\/em>) and \u03bd(<em>V<\/em>) are in \u03a9<em><sub>\u03bd<\/sub><\/em>, and since\u00a0<em>P<\/em> is prime,\u00a0\u03bd(<em>U<\/em>)\u00a0\u2229\u00a0\u03bd(<em>V<\/em>) = \u03bd(<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>) is not included in\u00a0\u03bd(<em>P<\/em>) either, so\u00a0<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em> is not included in\u00a0\u03bd(<em>P<\/em>) either, and therefore\u00a0\u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2013<em>P<\/em>)\u22600 by Lemma 4.<\/p>\n<p>In the converse direction, if\u00a0<em>P<\/em> satisfies the stated property, we consider any two elements\u00a0<em>U<\/em>,\u00a0<em>V<\/em>\u00a0of \u03a9<em><sub>\u03bd\u00a0<\/sub><\/em>such that <em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>\u00a0\u2286 \u03bd(<em>P<\/em>). \u00a0If neither\u00a0<em>U<\/em> nor\u00a0<em>V<\/em> is included in\u00a0\u03bd(<em>P<\/em>), then\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 and\u00a0\u03bc(<em>V\u2013<\/em><em>P<\/em>)\u22600 by Lemma 4. \u00a0Using the stated property,\u00a0\u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2013<em>P<\/em>)\u22600, so\u00a0<em>U<\/em>\u00a0\u2229\u00a0<em>V<\/em>\u00a0is not included in\u00a0\u03bd(<em>P<\/em>) either by Lemma 4: contradiction. \u00a0\u2610<\/p>\n<p>Proposition 5 sounds like irreducibility: the prime elements of \u03a9<em><sub>\u03bd\u00a0<\/sub><\/em>are the prime open subsets of\u00a0<em>X<\/em> &#8220;up to measure 0 sets&#8221; (whatever that means).<\/p>\n<p>At least, given a prime element\u00a0<em>P<\/em> of \u03a9<em><sub>\u03bd<\/sub><\/em>, Proposition 5 shows that the set\u00a0<em>F<\/em>(<em>P<\/em>) of open subsets\u00a0<em>U<\/em> of\u00a0<em>X<\/em>\u00a0such that\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 is a filter. \u00a0Since\u00a0\u03bc is Scott-continuous,\u00a0<em>F<\/em>(<em>P<\/em>) is Scott-open (you need to realize that\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 is equivalent to\u00a0\u03bc(<em>U<\/em>\u00a0\u222a\u00a0<em>P<\/em>)&gt;\u03bc(<em>P<\/em>) for that: use modularity). \u00a0The empty set is not in\u00a0<em>F<\/em>(<em>P<\/em>), and if\u00a0<em>U<\/em>\u00a0\u222a <em>V<\/em> is in\u00a0<em>F<\/em>(<em>P<\/em>) then\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)+\u03bc(<em>V<\/em>\u2013<em>P<\/em>) \u2265\u00a0\u03bc((<em>U<\/em>\u222a<em>V<\/em>)\u2013<em>P<\/em>) &gt; 0, so\u00a0<em>U<\/em> or\u00a0<em>V<\/em> is in\u00a0<em>F<\/em>(<em>P<\/em>). \u00a0It follows that\u00a0<em>F<\/em>(<em>P<\/em>) is a completely prime filter of open sets.<\/p>\n<p>If\u00a0<em>X<\/em> is sober, then that implies that there is a unique point\u00a0<em>x<\/em> in\u00a0<em>X<\/em> such that\u00a0<em>F<\/em>(<em>P<\/em>) is the collection of open neighborhoods of\u00a0<em>x<\/em>. \u00a0It follows that, for every open subset\u00a0<em>U<\/em> of\u00a0<em>X<\/em>,\u00a0<em>x<\/em> is in\u00a0<em>U<\/em> if and only if\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600. \u00a0Consider the open subset\u00a0<em>U<\/em><sub>0<\/sub> equal to the complement of\u00a0\u2193<em>x<\/em>. \u00a0Since\u00a0<em>x<\/em> is not in <em>U<\/em><sub>0<\/sub>,\u00a0\u03bc(<em>U<\/em><sub>0<\/sub>\u2013<em>P<\/em>)=0, or equivalently (by Lemma 4), <em>U<\/em><sub>0<\/sub>\u2286<em>\u03bd<\/em>(<em>P<\/em>)=<em>P<\/em>. \u00a0Conversely, since\u00a0\u03bc(<em>P<\/em>\u2013<em>P<\/em>)=0,\u00a0<em>x<\/em> is not in\u00a0<em>P<\/em>. \u00a0Since\u00a0<em>P<\/em> is upwards-closed, every element\u00a0<em>y<\/em>\u00a0of\u00a0<em>P<\/em> is in <em>U<\/em><sub>0<\/sub>, otherwise it would be below\u00a0<em>x<\/em>, and that would entail that\u00a0<em>x<\/em> is in\u00a0<em>P<\/em>. \u00a0Hence\u00a0<em>P<\/em> is included in <em>U<\/em><sub>0<\/sub>. \u00a0Together with <em>U<\/em><sub>0<\/sub>\u2286<em>P<\/em>, we obtain that <em>U<\/em><sub>0<\/sub>=<em>P<\/em>.<\/p>\n<p>Summing up, (if <em>X<\/em> is sober,)\u00a0<em>P<\/em> is the complement of\u00a0\u2193<em>x<\/em>, where\u00a0<em>x<\/em> is a point such that every open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em>\u00a0containing <em>P<\/em>\u00a0has strictly larger\u00a0\u03bc-measure than\u00a0<em>P<\/em>. \u00a0Let us call such a point <em>x<\/em>\u00a0a\u00a0\u03bc-<em>atom<\/em>: this is, roughly, a point that carries some\u00a0\u03bc-mass by itself.<\/p>\n<p>We have proved one half of:<\/p>\n<p><strong>Proposition 6.<\/strong> If\u00a0<em>X<\/em> is sober, then the prime elements\u00a0<em>P<\/em> of \u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0are\u00a0the complements of\u00a0\u2193<em>x<\/em>, where\u00a0<em>x<\/em> is a \u03bc-atom. \u00a0(More succinctly: the points of\u00a0\u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0are\u00a0the\u00a0\u03bc-atoms.)<\/p>\n<p>Proof. \u00a0It remains to show that if\u00a0<em>P<\/em> is the complement\u00a0of\u00a0\u2193<em>x<\/em>, where\u00a0<em>x<\/em> is a \u03bc-atom, then\u00a0<em>P<\/em> is a prime element of \u03a9<em><sub>\u03bd<\/sub><\/em>. \u00a0Let\u00a0<em>U<\/em> be any open subset of\u00a0<em>X<\/em> such that\u00a0<em>U<\/em>\u2261<em>P<\/em>. \u00a0If\u00a0<em>x<\/em> is in\u00a0<em>U<\/em>, then\u00a0\u03bc(<em>U<\/em>\u00a0\u222a\u00a0<em>P<\/em>)&gt;\u03bc(<em>U<\/em>) since\u00a0<em>x<\/em> is a \u03bc-atom. \u00a0However\u00a0<em>U<\/em>\u2261<em>P<\/em>\u00a0implies that\u00a0\u03bc(<em>U<\/em>\u00a0\u222a\u00a0<em>P<\/em>)=\u03bc(<em>U<\/em>\u00a0\u2229\u00a0<em>P<\/em>)\u2264\u03bc(<em>U<\/em>), so\u00a0<em>x<\/em> cannot be in\u00a0<em>U<\/em>. \u00a0By definition of\u00a0<em>P<\/em>,\u00a0<em>U<\/em> is then included in\u00a0<em>P<\/em>: for every\u00a0<em>y<\/em> in\u00a0<em>U<\/em>,\u00a0<em>y<\/em> cannot be below\u00a0<em>x<\/em> otherwise\u00a0<em>x<\/em> would be in\u00a0<em>U<\/em>, so\u00a0<em>y<\/em> is in\u00a0<em>P<\/em>. \u00a0We have just shown that\u00a0<em>P<\/em> is the largest open set in its\u00a0\u2261-equivalence class, namely that\u00a0<em>P<\/em>=<em>\u03bd<\/em>(<em>P<\/em>). \u00a0Therefore\u00a0<em>P<\/em> is in \u03a9<em><sub>\u03bd<\/sub><\/em>.<\/p>\n<p>Assume now, with the aim of applying Proposition 5, that we are given two open subsets\u00a0<em>U<\/em> and\u00a0<em>V<\/em> of\u00a0<em>X<\/em> such that\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>)\u22600 and\u00a0\u03bc(<em>V\u2013<\/em><em>P<\/em>)\u22600. \u00a0Then\u00a0<em>x<\/em> must be in\u00a0<em>U<\/em>, since otherwise\u00a0<em>U<\/em> would be included in\u00a0<em>P<\/em>, and therefore\u00a0\u03bc(<em>U<\/em>\u2013<em>P<\/em>) would be equal to 0. \u00a0Similarly,\u00a0<em>x<\/em> is in\u00a0<em>V<\/em>, hence\u00a0<em>x<\/em> is in <em>U<\/em>\u2229<em>V<\/em>. \u00a0Since\u00a0<em>x<\/em>\u00a0is a \u03bc-atom, \u03bc(<em>U<\/em>\u2229<em>V<\/em>\u2013<em>P<\/em>)\u22600. \u00a0Hence\u00a0<em>P<\/em> is prime. \u00a0\u2610<\/p>\n<p>Consider again Lebesgue measure\u00a0\u03bc on [0, 1]. \u00a0[0, 1], with its usual metric topology, is T<sub>2<\/sub>\u00a0hence sober, and we claim that there is no\u00a0\u03bc-atom. \u00a0Imagine there were one, call it\u00a0<em>x<\/em>. \u00a0The complement\u00a0<em>P<\/em> of\u00a0\u2193<em>x<\/em> is [0, x)\u00a0\u222a (x, 1], whose \u03bc-measure is 1,\u00a0and it is not true that every open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em> containing\u00a0<em>P<\/em> has strictly larger\u00a0\u03bc-measure: the only such\u00a0<em>U<\/em> is [0, 1], and it also has\u00a0\u03bc-measure equal to 1.<\/p>\n<p>Hence \u03a9<em><sub>\u03bd<\/sub><\/em>\u00a0simply has no points in this (important) case&#8230; but it is a very large sublocale. \u00a0Seen as a sieve, this is the collection of all crescents\u00a0<em>U<\/em>\u2013<em>V<\/em> of non-zero measure.<\/p>\n<ol>\n<li>Alex Simpson. \u00a0The locale of random sequences. \u00a0Invited talk, Continuity, Computability, Constructivity, From Logic to Algorithms Cologne, July 2009. \u00a0(Updated, November 19th, 2023: rather, see Alex Simpson, <a href=\"https:\/\/philpapers.org\/rec\/SIMMRA\">Measure, randomness and sublocales<\/a>, <em class=\"pubName\"><a class=\"discreet\" href=\"https:\/\/philpapers.org\/asearch.pl?pub=1339\">Annals of Pure and Applied Logic<\/a><\/em> 163(11):1642-1659, 2012.)<\/li>\n<li>Jorge Picado and Ale\u0161 Pultr. Frames and locales \u2014 topology without points. Birkh\u00e4user, 2010.<\/li>\n<li>Klaus Keimel and Jimmie Lawson. Measure extension theorems for T<sub>0<\/sub>-spaces. Topology and its Applications, 149(1\u20133), 57\u201383, 2005.<\/li>\n<\/ol>\n<p style=\"text-align: right;\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\" rel=\"attachment wp-att-993\">Jean Goubault-Larrecq<\/a>\u00a0(November 26th, 2018)<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-993 alignright\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" width=\"32\" height=\"44\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Alex Simpson has a lot of slides with very interesting ideas. \u00a0One of them is what he calls the locale of random sequences. \u00a0(Update, November 19th, 2022: this link now points nowhere. \u00a0I am unable to find his slides nowadays. &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1597\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-1597","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1597","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1597"}],"version-history":[{"count":20,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1597\/revisions"}],"predecessor-version":[{"id":5923,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1597\/revisions\/5923"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}