{"id":1342,"date":"2017-12-31T17:11:57","date_gmt":"2017-12-31T16:11:57","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1342"},"modified":"2022-11-19T15:18:43","modified_gmt":"2022-11-19T14:18:43","slug":"integration","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1342","title":{"rendered":"Integration"},"content":{"rendered":"

At the start of the book<\/a>, I had stated: “Topological convexity, topological measure theory, hyperspaces, and powerdomains will be treated in further volumes.” \u00a0The book<\/a> got out in 2013, but I wrote that in 2011,\u00a0almost seven years ago now. \u00a0What happened?<\/p>\n

Well, nothing went according to plan, but I in fact wrote plenty of things during the period. Let me tell you what happened… with a surprise in the middle\u00a0of the post.<\/p>\n

Version zero<\/h2>\n

I had been writing on semantic models for mixed probabilistic and non-deterministic choice, and one can still find one of the latest version of my notes<\/a> on the subject on my Web page. \u00a0But that was in French, and there were many things I wanted to do better.<\/p>\n

Version one<\/h2>\n

I had finished writing the non-Hausdorff topology book<\/a> in 2011, and I started my new project right away. \u00a0I started writing an introductory, motivating example from computer science… and developed it\u00a0at such a level of detail\u00a0that I had written more than 50 pages and the book had not even started. \u00a0Hence I decided to scrap that version.<\/p>\n

Version two<\/h2>\n

I decided I should start all over again. \u00a0Since I needed a notion of integral, and Choquet integration, which has my preference, is based on the Riemann integral of functions on the real line R<\/strong>, I decided to start with the Riemann integral on R<\/strong>. \u00a0However, the Riemann integral is perhaps the worst notion of integral in existence. \u00a0So I decided to talk about the Kurzweil-Henstock integral, which is a kind of miracle (see below). \u00a0Now, by the time I had finished talking about that, I already had more than 150 pages… and again I had not started touching the real subject of the new book.<\/p>\n

I also realized at about that time that there were many books on the subject, including one that had come out during the same period. \u00a0Hence I decided to abandon that version as well. \u00a0However, I preferred to push it to completion. \u00a0But I could not decently publish it\u2014not new enough. \u00a0I kept it, not knowing what to do with it.<\/p>\n

This is the surprise I promised you: that book is here<\/a>. \u00a0(Login: guest, password: guest.) \u00a0Happy New Year 2018!<\/p>\n

Version three<\/h2>\n

In the meantime, in 2011 or 2012, I realized that I should write the new book\u00a0with Klaus\u00a0Keimel, who was one of the best experts on the subject. He agreed,\u00a0and it took me three years to send him a possible outline. We\u00a0discussed the contents of the book in early 2015, but nothing really\u00a0got started for some time: I had more urgent things to do, then he had\u00a0to finish a paper, then I did not have time, and so on… until Klaus suggested we met for a week in Schloss\u00a0Dagstuhl, away from all constraints. We finally started it there, in\u00a0February 2016. However, as soon as we returned home, time escaped me\u00a0again. \u00a0Also, I had refereed Klaus’s paper and therefore given him some more work to do before he could come back to the book. \u00a0As a result, the book had not advanced much until Klaus’s sad demise on\u00a0November 18th, 2017.<\/p>\n

I owe a lot to Klaus, and I decided I should give myself a good kick somewhere and start all over again. \u00a0Let us hope that version four<\/strong> will be the right one.<\/p>\n

The Kurzweil-Henstock integral<\/h2>\n

The Riemann integral of a function f<\/em> : [a<\/em>,\u00a0b<\/em>]\u00a0\u2192\u00a0R<\/strong>\u00a0is defined as follows. \u00a0We subdivide the interval\u00a0[a<\/em>,\u00a0b<\/em>] by cutting it at points a<\/em>0<\/sub>=a<\/em> \u2264 a<\/em>1<\/sub> \u2264 a<\/em>1<\/sub> \u2264 … \u2264 an-1<\/sub><\/em> \u2264 an<\/sub><\/em>=b<\/em>, pick points t<\/i>1<\/sub>\u00a0in [a<\/em>0<\/sub>,\u00a0a<\/em>1<\/sub>], t<\/i>2<\/sub>\u00a0in [a<\/em>1<\/sub>,\u00a0a<\/em>2<\/sub>], …, and tn<\/sub><\/em>\u00a0in [an-1<\/sub><\/em>,\u00a0an<\/sub><\/em>]. \u00a0That data is called a\u00a0pointed subdivision<\/em> of [a<\/em>,\u00a0b<\/em>]. \u00a0For such a pointed subdivision\u00a0D<\/em>, the\u00a0Riemann sum<\/em> of\u00a0f<\/em> relatively to\u00a0D<\/em> is\u00a0\u2211i=1<\/sub>n<\/sup><\/em> (ai<\/sub><\/em>\u2014ai-1<\/sub><\/em>) f<\/em> (ti<\/sub><\/em>). \u00a0Let us write that as\u00a0\u222bD<\/sub><\/em> f<\/em>. \u00a0If\u00a0\u222bD<\/sub><\/em> f<\/em> tends to some number when the width of the intervals\u00a0[ai-1<\/sub><\/em>,\u00a0ai<\/sub><\/em>] tends to 0, then the result if the\u00a0Riemann integral<\/em> of\u00a0f<\/em> on the interval [a<\/em>,\u00a0b<\/em>].<\/p>\n

Formally, we can cast that as the limit of a net, as follows. \u00a0The indexing preorder is given by pairs (D<\/em>,\u00a0\u03b7) where\u00a0D<\/em> is a pointed subdivision as above and such that ai<\/sub><\/em>\u2014ai-1<\/sub><\/em><\u03b7 for every\u00a0i<\/em>. \u00a0Moreover, we say that\u00a0(D<\/em>,\u00a0\u03b7)\u00a0\u2aaf\u00a0(D’<\/em>,\u00a0\u03b7’) if\u00a0\u03b7\u2265\u03b7’. \u00a0In other words, we are looking at what happens when the integration step\u00a0\u03b7 goes to 0. \u00a0We form a net by saying that its element at index\u00a0(D<\/em>,\u00a0\u03b7) is \u222bD<\/sub><\/em> f<\/em>, and define the Riemann integral as the limit of that net, if the limit exists.<\/p>\n

There are other ways of defining the Riemann integral. \u00a0Some of you may have heard about Darboux sums\u2014a simpler construction\u2014but that only works to define the Riemann integral of continuous maps. \u00a0There are also slightly simpler ways to define the Riemann integral itself, for example by restricting to pointed subdivisions where each interval\u00a0[ai-1<\/sub><\/em>,\u00a0ai<\/sub><\/em>] has exactly the same length.<\/p>\n

However, the point is that there is a simple modification of this modification, the Kurzweil-Henstock integral, which has much better properties than the Riemann integral.<\/p>\n

What we do is replace the parameter\u00a0\u03b7, which bounds the width of subintervals\u00a0[ai-1<\/sub><\/em>,\u00a0ai<\/sub><\/em>]\u00a0uniformly<\/em>, by a so-called\u00a0gauge<\/em>\u00a0\u03b4, which is a map which says what the allowed width of the interval is around each point ti<\/sub><\/em>.<\/p>\n

Formally, a gauge is a map\u00a0\u03b4 :\u00a0[a<\/em>,\u00a0b<\/em>]\u00a0\u2192 (0,\u00a0\u221e). \u00a0A pointed subdivision\u00a0D<\/em> is\u00a0\u03b4-fine<\/em> if and only if\u00a0ai<\/sub><\/em>\u2014ai-1<\/sub><\/em><\u03b4(ti<\/sub><\/em>) for every\u00a0i<\/em>. \u00a0We now define a net whose index set\u00a0consists of pairs (D<\/em>, \u03b4) where\u00a0D<\/em> is a\u00a0\u03b4-fine pointed subdivision, and whose element at index\u00a0(D<\/em>, \u03b4) is again\u00a0\u222bD<\/sub><\/em> f<\/em>. \u00a0This time, we preorder the index set by\u00a0(D<\/em>, \u03b4)\u00a0\u2aaf\u00a0(D’<\/em>, \u03b4’) if \u03b4(t<\/em>)\u2265\u03b4'(t<\/em>) for every t<\/em> in\u00a0[a<\/em>,\u00a0b<\/em>]. \u00a0The limit of that net, if it exists, is the\u00a0Kurzweil-Henstock integral<\/em> of\u00a0f<\/em> on\u00a0[a<\/em>,\u00a0b<\/em>]. \u00a0We write\u00a0\u222ba<\/sub>b<\/sup><\/em> f<\/em> for that integral.<\/p>\n

The only difference with the Riemann integral is that we measure how fine a pointed subdivision is by resorting to a gauge, which\u00a0is allowed to give varying widths of intervals across\u00a0[a<\/em>,\u00a0b<\/em>]. \u00a0But that makes a huge difference! \u00a0Let us list a few:<\/p>\n