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2612:. Currently the article has two self-contained sections about two relatively distinct subjects, both of which would seem to be worthwhile subjects for articles of their own. There seems to be no reason for this other than inertia. The current referencing style makes in unclear which source corresponds to which subject, so the references would reviewed to see which article they should be placed in.--
22:
2507:"relationship between differentiation and integration is commonly characterized in the framework of Riemann integration" — I think, it means that, having a continuously differentiable function, you can restore it from its derivative by Riemann integration. Poor English? Do you prefer "described" instead of "characterized"? Propose your formulation.
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Today, I made what should have been a very small (one character) edit to the bit on R-N theorem. I apologise that it took me 4 attempts to get mu and nu the correct way around in the text of my accompanying edit summary. Hence the multiple undos and redos whose sole purpose was to correct the edit
2627:
No, I believe, these two notions are very much related (as written in Sect.2.1). The sources could be better; I'll add more sources soon. And they will be sources that do describe both subjects (and their relation). True, the relation becomes weak when we turn to functions with values in metric
3121:
The
Generalizations section discusses functions whose range is a general metric space, but the domain is still the real line. Is there a standard way to generalize absolute continuity to functions whose domain is a general metric space, or at least a multi-dimensional space? E.g., when does a
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f )? Or it should be in a different article? And what about ACG and ACG* functions? Probably this would require considering absolute continuity of functions on an arbitrary set E in R, instead of an interval.
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2884:; you get the divergent harmonic series. And the derivative of this function is not integrable. On the other hand, it is uniformly continuous, since it can be extended by continuity to . Also, the function
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I am not too familiar with absolute continuity but it seems a little odd that it should be stronger than
Lipschitz, as the lead paragraph currently claims. Could someone add a clarification?
2467:
This page ought to have some mention of the
Lebesgue version of the Fundamental Theorem of Calculus. To my mind that's a major reason for being interested in absolutely continuous functions.
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0 such that if a set A satisfies μ(A)< δ then it satisfies ν(A) < ε. the (<=) part is obvious. some finiteness assumption seems to be needed on ν, for a short proof of the converse:
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2691:"(These examples are continuous but not uniformly continuous.)" — True. but somewhat misleading: a uniformly continuous function (even on ) need not be absolutely continuous (try the
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737:{\displaystyle \mu :{\mathcal {B}}\rightarrow \left,A\mapsto {\begin{cases}0,&{\text{if }}\lambda \left(A\right)=0\\\infty ,&{\text{if }}\lambda \left(A\right): -->
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If we are restricting ourselves to compact intervals of the real line then uniformly continuous is equivalent to continuous, correct? What is this trying to say here?
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1756:{\displaystyle \infty =\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)\geq \sum _{n=1}^{\infty }\mu \left(A_{n}\right)\geq \mu \left(A_{i_{0}}\right)=\infty ,}
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Really, here "locally" is not needed, since the measure is assumed to be finite. (Otherwise, "locally" is needed, and means: on every bounded interval.)
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Is the sentence "This relationship is commonly characterized ...." in the first paragraph correct (English)? I can't understand what does it means.
3325:"; and I do not know how to stop this process. (Similarly, "die" and "dice" are changed repeatedly, back and forth, in some probability articles.)
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is better. I suspect (but do not know) that there are examples to show that some (all?) of the other inclusions given in the lead are strict too.
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I'd say we're missing the alternative characterisation of absolute continuity of measures here, the epsilon-delta one... Anyone can put it on?
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1495:{\displaystyle \lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\lambda \left(A_{n}\right)=\sum _{n=1}^{\infty }0=0,}
3157:
a straighforward thought is that a cumulative distribution function of an absolutely continuous measure fits. But still, I did not see it.
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spaces. This case could be treated in a separate section, for convenience of readers that are interested only in real-valued functions.
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Should I add some information on so-called AC* functions (absolutely continuous in the narrow sense)? (in definition the value |f(x
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1221:{\displaystyle \sum _{n=1}^{\infty }\mu \left(A_{n}\right)=\sum _{n=1}^{\infty }0=0=\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right).}
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Yeah. It's interestin' to think that no matter how small δ is, you can find an infinite amount of "variation" in the function
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or the like rather than nu(A)<epsilon.) Maybe I'll put it there if you don't, after I check a couple of sources....)
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180:. However, is it not absolutely continuous, as the Cantor distribution is not absolutely continuous with respect to
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Hi Boris. What you say is correct. I was tryin' to say that these are examples which are not absolutely continuous
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And no wonder. Given a divergent series, no matter how far is its tail, you can find an infinite sum in the tail.
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And by the way, the "Equivalent definitions" are still equivalent on a bounded (but maybe not closed) interval.
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176:, when restricted to the compact interval , is a continuous function defined on a compact set, and is therefore
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The explanation in the subsection "Relation between the two notions of absolute continuity" uses the term "
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uniformly continuous but not absolutely continuous. But here's a question for you: What about the function
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220:, not for measures. The one I mean is of the sort of: mu is abs. cont. w.r.t. nu if for every epsilon: -->
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1031:{\displaystyle 0\leq \lambda \left(A_{i}\right)\leq \lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=0}
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About the chain of inclusions I wonder: what is meant by differentiability of a function over a compact
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they are not uniformly continuous, but that there are other functions (like the Cantor function) that
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2180:{\displaystyle A_{\delta }:=\left(-{\frac {\delta }{4}},{\frac {\delta }{4}}\right)\in {\mathcal {B}}}
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absolutely continuous", but this term was not defined before. Can someone please add a definition? --
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This makes sense, but you probably know better than I do. I just copied the chain from elsewhere. --
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As far as I remember, various editors change these things repeatedly, back and forth, here and in "
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3213:"We have the following chains of inclusions for functions over a compact subset of the real line:
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on the domain (0, 1] is not absolutely continuous; moreover, it is not of bounded variation. Try
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P.S. In fact, I see that under "properties" it says the opposite. The lead should be corrected.
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What we have to assume at least to make the statement true (and your proof work), is that
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Is there an example of a uniformly continuous function that is not absolutely continous?
865:{\displaystyle \left(A_{n}\right)_{n\in \mathbb {N} }\in {\mathcal {B}}^{\mathbb {N} }}
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Two books are added. I'll also add some inline citations (but not for metric spaces).
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2243:{\displaystyle 0<\lambda \left(A_{\delta }\right)={\frac {\delta }{2}}<\delta }
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0 such that if a set A satisfies mu(A)<delta then it satisfies nu(A)<epsilon.
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suppose the ε-δ condition doesn't hold. so we have some ε and a sequence of sets A
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Something strange happened to the definition in the last two edits. Reverted.
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in the domain (0, 1]? Is it uniformly continuous? Absolutely continuous?
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I don't see it! There is an epsilon-delta definiton, but that one is for
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Am I correct in assuming that this interval does not have to be finite?
2299:{\displaystyle \mu \left(A_{\delta }\right)=\infty \geq 1=\varepsilon }
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Furthermore, by definition of absolute continuity and by definition of
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934:{\displaystyle \lambda \left(\bigcup _{n=1}^{\infty }A_{n}\right)=0}
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It is uniformly continuous but not absolutely continuous. So what?
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I never saw such a generalization for metric space domain. For
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You are right; I was not careful. Two corrections are made.
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Your proof only works if you have finiteness assumptions on
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on the domain (0, 1] is absolutely continuous whenever
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be a sequence of disjoint sets. We consider two cases:
505:{\displaystyle \lim _{n\rightarrow \infty }\nu (B_{n})}
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Oh, OK. Go ahead and put it in. (But write either ν(
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is meant (here and "Lipschitz continuous" as well)?
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you probably mean ν << μ iff for every ε : -->
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3078:Undefined term: "locally absolutely continuous"
3117:Absolute continuity in more than one dimension
3244:Yes, you are right. And the same happens on
1360:-measure zero, since otherwise we would have
796:{\displaystyle \mu \left(\emptyset \right)=0}
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590:{\displaystyle \mu :=\infty \cdot \lambda }
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266:{\displaystyle \nu (A)<\varepsilon \,}
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205:That's not missing; it's in the article.
2660:Explanation re a small edit on 7th March
1861:{\displaystyle \lambda \left(A\right)=0}
1333:{\displaystyle A_{i},i\in \mathbb {N} }
261:
49:
19:
2378:{\displaystyle \lambda (A)<\infty }
1591:. Then we have again by definition of
541:be Lebesgue measure on the Borel-sets
414:{\displaystyle \nu (B_{n})<\infty }
352:, because only if at least one of the
2877:{\displaystyle x_{n}=1/((n+0.5)\pi )}
2779:OK with "because". Now, the function
1530:{\displaystyle i_{0}\in \mathbb {N} }
7:
3138:considered absolutely continuous? --
2604:This should really be two articles:
1824:{\displaystyle \mu \left(A\right)=0}
95:This article is within the scope of
1964:{\displaystyle \varepsilon :=1: -->
1891:{\displaystyle A\in {\mathcal {B}}}
453:{\displaystyle \nu (\bigcap B_{n})}
38:It is of interest to the following
3166:of the real line? Maybe a compact
2385:or something in that direction. --
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2343:{\displaystyle \mu (A)<\infty }
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3363:Mid-priority mathematics articles
1059:{\displaystyle i\in \mathbb {N} }
301:ε. Take the decreasing sequence B
115:Knowledge:WikiProject Mathematics
2960:{\displaystyle \varepsilon : -->
2687:Absolute continuity of functions
317:) = 0 but, if ν is finite, ν(∩ B
118:Template:WikiProject Mathematics
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135:This article has been rated as
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2622:20:11, 18 February 2010 (UTC)
2610:Absolutely continuous measure
2595:04:19, 22 November 2009 (UTC)
2517:11:37, 15 December 2019 (UTC)
2501:07:51, 15 December 2019 (UTC)
2477:01:58, 4 September 2008 (UTC)
2462:09:25, 14 February 2008 (UTC)
1938:. We will now prove that for
766:is indeed a measure. Clearly
296:) < ∞ and for every n, ν(A
109:and see a list of open tasks.
3358:C-Class mathematics articles
3266:What about Cantor function?
2551:12:11, 27 October 2008 (UTC)
2537:12:09, 27 October 2008 (UTC)
3198:17:24, 25 August 2014 (UTC)
3180:20:59, 23 August 2014 (UTC)
3148:20:01, 23 August 2014 (UTC)
3111:15:18, 22 August 2014 (UTC)
3096:14:28, 22 August 2014 (UTC)
3035:{\displaystyle x\sin(1/x).}
2102:{\displaystyle \delta : -->
1993:{\displaystyle \delta : -->
189:18:10, 2 January 2007 (UTC)
167:17:43, 2 January 2007 (UTC)
3379:
3258:15:32, 28 April 2015 (UTC)
2815:{\displaystyle x\sin(1/x)}
2758:{\displaystyle x\sin(1/x)}
2417:16:30, 11 April 2010 (UTC)
2395:09:19, 11 April 2010 (UTC)
3335:15:43, 16 June 2016 (UTC)
3317:14:16, 16 June 2016 (UTC)
2681:21:25, 7 March 2013 (UTC)
2654:20:28, 3 March 2010 (UTC)
2638:20:11, 3 March 2010 (UTC)
2570:08:27, 18 June 2009 (UTC)
2431:09:23, 25 July 2007 (UTC)
2426:is finiteness necessary?
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67:
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3279:19:27, 3 July 2015 (UTC)
3066:08:15, 6 July 2014 (UTC)
3052:06:15, 6 July 2014 (UTC)
3042:in the interval (0, δ)!
2992:17:10, 4 July 2014 (UTC)
2978:16:55, 4 July 2014 (UTC)
2775:12:41, 4 July 2014 (UTC)
2705:09:07, 3 July 2014 (UTC)
2111:be arbitrary and choose
1931:{\displaystyle \lambda }
1763:and therefore equalitiy.
1502:a contradiction. So let
1353:{\displaystyle \lambda }
534:{\displaystyle \lambda }
278:22:29, 5 June 2007 (UTC)
227:22:21, 5 June 2007 (UTC)
210:01:25, 31 May 2007 (UTC)
200:01:21, 31 May 2007 (UTC)
141:project's priority scale
746:. We shall prove, that
744:0.\end{cases}}}" /: -->
221:0 there is a delta: -->
98:WikiProject Mathematics
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372:{\displaystyle B_{n}}
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3225:uniformly continuous
3210:It currently reads:
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178:uniformly continuous
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2107:
2106:
2100:
2081:
2079:
2078:
2073:
2065:
2044:
2042:
2041:
2036:
2022:
2001:
1998:
1997:
1991:
1972:
1969:
1968:
1962:
1937:
1935:
1934:
1929:
1917:
1915:
1914:
1909:
1897:
1895:
1894:
1889:
1887:
1886:
1867:
1865:
1864:
1859:
1851:
1830:
1828:
1827:
1822:
1814:
1793:
1791:
1790:
1785:
1762:
1760:
1759:
1754:
1743:
1739:
1738:
1737:
1736:
1712:
1708:
1707:
1690:
1685:
1667:
1663:
1662:
1661:
1651:
1646:
1610:
1608:
1607:
1602:
1590:
1587:
1586:
1580:
1572:
1568:
1567:
1566:
1565:
1536:
1534:
1533:
1528:
1526:
1518:
1517:
1501:
1499:
1498:
1493:
1478:
1473:
1455:
1451:
1450:
1433:
1428:
1410:
1406:
1405:
1404:
1394:
1389:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1329:
1315:
1314:
1298:
1295:
1294:
1288:
1280:
1276:
1275:
1274:
1264:
1259:
1227:
1225:
1224:
1219:
1214:
1210:
1209:
1208:
1198:
1193:
1157:
1152:
1134:
1130:
1129:
1112:
1107:
1085:
1083:
1082:
1077:
1065:
1063:
1062:
1057:
1055:
1037:
1035:
1034:
1029:
1021:
1017:
1016:
1015:
1005:
1000:
974:
970:
969:
940:
938:
937:
932:
924:
920:
919:
918:
908:
903:
871:
869:
868:
863:
861:
860:
859:
853:
852:
842:
841:
840:
828:
824:
823:
802:
800:
799:
794:
786:
765:
763:
762:
757:
745:
742:
741:
735:
733:
732:
720:
706:
703:
683:
669:
666:
639:
635:
617:
616:
596:
594:
593:
588:
564:
562:
561:
556:
554:
553:
540:
538:
537:
532:
511:
509:
508:
503:
498:
497:
481:
459:
457:
456:
451:
446:
445:
420:
418:
417:
412:
401:
400:
378:
376:
375:
370:
368:
367:
351:
349:
348:
343:
313:... . Then μ(∩ B
272:
270:
269:
264:
182:Lebesgue measure
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
3378:
3377:
3373:
3372:
3371:
3369:
3368:
3367:
3348:
3347:
3327:Boris Tsirelson
3271:Boris Tsirelson
3250:Boris Tsirelson
3208:
3172:Boris Tsirelson
3119:
3103:Boris Tsirelson
3080:
3058:Boris Tsirelson
2998:
2997:
2984:Boris Tsirelson
2970:Boris Tsirelson
2942:
2941:
2891:
2886:
2885:
2829:
2824:
2823:
2781:
2780:
2724:
2723:
2697:Boris Tsirelson
2693:Cantor function
2689:
2666:
2662:
2646:Boris Tsirelson
2630:Boris Tsirelson
2602:
2581:
2562:Boris Tsirelson
2558:
2525:
2509:Boris Tsirelson
2450:
2447:
2443:
2409:Boris Tsirelson
2352:
2351:
2317:
2316:
2264:
2260:
2252:
2251:
2207:
2203:
2189:
2188:
2135:
2131:
2118:
2113:
2112:
2084:
2083:
2055:
2047:
2046:
2012:
2004:
2003:
1975:
1974:
1940:
1939:
1920:
1919:
1900:
1899:
1898:), we see that
1870:
1869:
1841:
1833:
1832:
1804:
1796:
1795:
1776:
1775:
1728:
1723:
1719:
1699:
1695:
1653:
1631:
1627:
1613:
1612:
1593:
1592:
1557:
1552:
1548:
1539:
1538:
1509:
1504:
1503:
1442:
1438:
1396:
1374:
1370:
1362:
1361:
1342:
1341:
1306:
1301:
1300:
1299:. Then not all
1266:
1244:
1240:
1231:
1230:
1200:
1178:
1174:
1121:
1117:
1088:
1087:
1068:
1067:
1040:
1039:
1007:
985:
981:
961:
957:
943:
942:
941:. Then we have
910:
888:
884:
876:
875:
846:
815:
811:
810:
805:
804:
776:
768:
767:
748:
747:
738:0.\end{cases}}}
728:
727:
710:
700:
691:
690:
673:
663:
650:
625:
621:
599:
598:
567:
566:
543:
542:
523:
522:
489:
462:
461:
437:
423:
422:
392:
381:
380:
359:
354:
353:
334:
333:
324:
320:
316:
312:
308:
304:
299:
295:
291:
238:
237:
174:Cantor function
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3376:
3374:
3366:
3365:
3360:
3350:
3349:
3346:
3345:
3344:
3343:
3342:
3341:
3340:
3339:
3338:
3337:
3305:
3304:
3303:
3261:
3260:
3235:
3234:
3207:
3204:
3203:
3202:
3201:
3200:
3183:
3182:
3159:
3158:
3118:
3115:
3114:
3113:
3079:
3076:
3075:
3074:
3073:
3072:
3071:
3070:
3069:
3068:
3031:
3028:
3025:
3021:
3017:
3014:
3011:
3008:
3005:
2980:
2956:
2953:
2950:
2929:
2926:
2922:
2918:
2915:
2912:
2909:
2904:
2901:
2898:
2894:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2848:
2844:
2841:
2836:
2832:
2811:
2808:
2804:
2800:
2797:
2794:
2791:
2788:
2754:
2751:
2747:
2743:
2740:
2737:
2734:
2731:
2688:
2685:
2661:
2658:
2657:
2656:
2641:
2640:
2601:
2598:
2580:
2573:
2557:
2554:
2524:
2521:
2520:
2519:
2483:
2481:
2465:
2449:
2445:
2441:
2439:
2437:
2435:
2424:
2423:
2422:
2421:
2420:
2419:
2400:
2399:
2398:
2397:
2374:
2371:
2368:
2365:
2362:
2359:
2339:
2336:
2333:
2330:
2327:
2324:
2310:
2309:
2308:
2307:
2295:
2292:
2289:
2286:
2283:
2280:
2276:
2271:
2267:
2263:
2259:
2239:
2236:
2231:
2228:
2223:
2219:
2214:
2210:
2206:
2202:
2199:
2196:
2174:
2169:
2165:
2159:
2156:
2151:
2146:
2143:
2138:
2134:
2130:
2125:
2121:
2098:
2095:
2092:
2071:
2068:
2064:
2061:
2058:
2054:
2034:
2031:
2028:
2025:
2021:
2018:
2015:
2011:
1989:
1986:
1983:
1960:
1957:
1954:
1951:
1948:
1927:
1907:
1885:
1880:
1877:
1857:
1854:
1850:
1847:
1844:
1840:
1820:
1817:
1813:
1810:
1807:
1803:
1783:
1769:
1768:
1767:
1766:
1765:
1764:
1752:
1749:
1746:
1742:
1735:
1731:
1726:
1722:
1718:
1715:
1711:
1706:
1702:
1698:
1694:
1689:
1684:
1681:
1678:
1674:
1670:
1666:
1660:
1656:
1650:
1645:
1642:
1639:
1635:
1630:
1626:
1623:
1620:
1600:
1578:
1575:
1571:
1564:
1560:
1555:
1551:
1547:
1525:
1521:
1516:
1512:
1491:
1488:
1485:
1482:
1477:
1472:
1469:
1466:
1462:
1458:
1454:
1449:
1445:
1441:
1437:
1432:
1427:
1424:
1421:
1417:
1413:
1409:
1403:
1399:
1393:
1388:
1385:
1382:
1378:
1373:
1369:
1349:
1328:
1324:
1321:
1318:
1313:
1309:
1286:
1283:
1279:
1273:
1269:
1263:
1258:
1255:
1252:
1248:
1243:
1239:
1228:
1217:
1213:
1207:
1203:
1197:
1192:
1189:
1186:
1182:
1177:
1173:
1170:
1167:
1164:
1161:
1156:
1151:
1148:
1145:
1141:
1137:
1133:
1128:
1124:
1120:
1116:
1111:
1106:
1103:
1100:
1096:
1075:
1054:
1050:
1047:
1027:
1024:
1020:
1014:
1010:
1004:
999:
996:
993:
989:
984:
980:
977:
973:
968:
964:
960:
956:
953:
950:
930:
927:
923:
917:
913:
907:
902:
899:
896:
892:
887:
883:
858:
851:
845:
839:
835:
832:
827:
822:
818:
814:
792:
789:
785:
782:
779:
775:
755:
731:
726:
723:
719:
716:
713:
709:
701:
699:
696:
693:
692:
689:
686:
682:
679:
676:
672:
664:
662:
659:
656:
655:
653:
648:
645:
642:
638:
634:
631:
628:
624:
620:
615:
610:
607:
586:
583:
580:
577:
574:
552:
530:
516:
515:
514:
513:
501:
496:
492:
488:
485:
480:
477:
474:
470:
449:
444:
440:
436:
433:
430:
410:
407:
404:
399:
395:
391:
388:
366:
362:
341:
327:
326:
322:
318:
314:
310:
306:
302:
297:
293:
289:
260:
257:
254:
251:
248:
245:
230:
229:
213:
212:
197:189.177.62.204
193:
192:
191:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3375:
3364:
3361:
3359:
3356:
3355:
3353:
3336:
3332:
3328:
3324:
3320:
3319:
3318:
3314:
3310:
3306:
3302:
3301:
3296:
3295:
3290:
3289:
3285:
3284:
3282:
3281:
3280:
3276:
3272:
3268:
3267:
3265:
3264:
3263:
3262:
3259:
3255:
3251:
3247:
3243:
3242:
3241:
3238:
3233:
3232:
3227:
3226:
3221:
3220:
3216:
3215:
3214:
3211:
3205:
3199:
3195:
3191:
3187:
3186:
3185:
3184:
3181:
3177:
3173:
3169:
3165:
3161:
3160:
3156:
3152:
3151:
3150:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3116:
3112:
3108:
3104:
3100:
3099:
3098:
3097:
3093:
3089:
3085:
3077:
3067:
3063:
3059:
3055:
3054:
3053:
3049:
3045:
3029:
3023:
3019:
3015:
3009:
3006:
3003:
2995:
2994:
2993:
2989:
2985:
2981:
2979:
2975:
2971:
2954:
2951:
2948:
2924:
2920:
2916:
2910:
2907:
2902:
2899:
2896:
2892:
2868:
2862:
2859:
2856:
2846:
2842:
2839:
2834:
2830:
2806:
2802:
2798:
2792:
2789:
2786:
2778:
2777:
2776:
2772:
2768:
2749:
2745:
2741:
2735:
2732:
2729:
2721:
2720:
2715:
2714:
2709:
2708:
2707:
2706:
2702:
2698:
2694:
2686:
2684:
2682:
2678:
2674:
2670:
2659:
2655:
2651:
2647:
2643:
2642:
2639:
2635:
2631:
2626:
2625:
2624:
2623:
2619:
2615:
2611:
2607:
2599:
2597:
2596:
2592:
2588:
2584:
2578:
2574:
2572:
2571:
2567:
2563:
2555:
2553:
2552:
2548:
2544:
2539:
2538:
2534:
2530:
2522:
2518:
2514:
2510:
2506:
2505:
2504:
2502:
2498:
2494:
2490:
2482:
2479:
2478:
2474:
2470:
2464:
2463:
2459:
2455:
2436:
2433:
2432:
2429:
2418:
2414:
2410:
2406:
2405:
2404:
2403:
2402:
2401:
2396:
2392:
2388:
2369:
2363:
2357:
2334:
2328:
2322:
2314:
2313:
2312:
2311:
2293:
2290:
2287:
2284:
2278:
2274:
2269:
2265:
2261:
2257:
2237:
2234:
2229:
2226:
2221:
2217:
2212:
2208:
2204:
2200:
2197:
2194:
2167:
2163:
2157:
2154:
2149:
2144:
2141:
2136:
2132:
2128:
2123:
2119:
2096:
2093:
2090:
2069:
2066:
2062:
2059:
2056:
2052:
2032:
2029:
2026:
2023:
2019:
2016:
2013:
2009:
1987:
1984:
1981:
1958:
1955:
1952:
1949:
1946:
1925:
1905:
1878:
1875:
1855:
1852:
1848:
1845:
1842:
1838:
1818:
1815:
1811:
1808:
1805:
1801:
1781:
1773:
1772:
1771:
1770:
1750:
1744:
1740:
1733:
1729:
1724:
1720:
1716:
1713:
1709:
1704:
1700:
1696:
1692:
1682:
1679:
1676:
1672:
1668:
1664:
1658:
1654:
1643:
1640:
1637:
1633:
1628:
1624:
1621:
1598:
1576:
1573:
1569:
1562:
1558:
1553:
1549:
1545:
1519:
1514:
1510:
1489:
1486:
1483:
1480:
1470:
1467:
1464:
1460:
1456:
1452:
1447:
1443:
1439:
1435:
1425:
1422:
1419:
1415:
1411:
1407:
1401:
1397:
1386:
1383:
1380:
1376:
1371:
1367:
1347:
1322:
1319:
1316:
1311:
1307:
1284:
1281:
1277:
1271:
1267:
1256:
1253:
1250:
1246:
1241:
1237:
1229:
1215:
1211:
1205:
1201:
1190:
1187:
1184:
1180:
1175:
1171:
1168:
1165:
1162:
1159:
1149:
1146:
1143:
1139:
1135:
1131:
1126:
1122:
1118:
1114:
1104:
1101:
1098:
1094:
1073:
1048:
1045:
1025:
1022:
1018:
1012:
1008:
997:
994:
991:
987:
982:
978:
975:
971:
966:
962:
958:
954:
951:
948:
928:
925:
921:
915:
911:
900:
897:
894:
890:
885:
881:
874:
873:
843:
833:
830:
825:
820:
816:
812:
790:
787:
783:
777:
773:
753:
724:
721:
717:
714:
711:
707:
697:
687:
684:
680:
677:
674:
670:
660:
657:
651:
643:
640:
636:
629:
626:
622:
608:
605:
584:
581:
575:
572:
528:
520:
519:
518:
517:
494:
490:
483:
472:
442:
438:
434:
428:
405:
397:
393:
386:
364:
360:
339:
331:
330:
329:
328:
287:
286:
285:
280:
279:
276:
275:Michael Hardy
258:
255:
249:
243:
235:
228:
225:
224:189.177.58.19
219:
215:
214:
211:
208:
207:Michael Hardy
204:
203:
202:
201:
198:
190:
187:
183:
179:
175:
171:
170:
169:
168:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
3298:
3292:
3286:
3239:
3236:
3229:
3223:
3217:
3212:
3209:
3167:
3163:
3154:
3135:
3131:
3127:
3123:
3120:
3083:
3081:
3044:Eric Kvaalen
2767:Eric Kvaalen
2718:
2717:
2712:
2711:
2690:
2673:138.40.68.40
2667:— Preceding
2663:
2603:
2585:
2582:
2576:
2559:
2540:
2526:
2493:82.136.67.75
2487:— Preceding
2484:
2480:
2469:72.25.102.62
2466:
2438:
2434:
2425:
1973:there is no
281:
236:) < ε or
233:
231:
217:
194:
186:Sullivan.t.j
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
2967:0.}" /: -->
2454:a_dergachev
321:) = lim ν(B
292:where ∑ μ(A
112:Mathematics
103:mathematics
59:Mathematics
3352:Categories
3300:continuous
3231:continuous
3190:Erel Segal
3140:Erel Segal
3088:Erel Segal
2109:0}" /: -->
2002:such that
2000:0}" /: -->
1971:0}" /: -->
1589:0}" /: -->
1340:can be of
1297:0}" /: -->
803:. Now let
379:satisfies
3122:function
2944:0.}": -->
2575:Interval
2556:Reverting
2350:whenever
1794:(we have
1038:for alle
218:functions
172:Yes. The
3283:Perhaps
3168:interval
2669:unsigned
2587:TomyDuby
2489:unsigned
2387:Phoemuex
2086:0}": -->
1977:0}": -->
1942:0}": -->
1541:0}": -->
1233:0}": -->
3309:YohanN7
3084:locally
2713:because
2543:Katzmik
2529:Katzmik
2428:Mct mht
2187:. Then
300:) : -->
164:Albmont
139:on the
30:C-class
3164:subset
2614:RDBury
2250:, but
325:) ≥ ε.
36:scale.
3134:) on
2952:: -->
2600:Split
2444:)-f(y
2094:: -->
1985:: -->
1956:: -->
1574:: -->
1537:with
1282:: -->
722:: -->
3331:talk
3313:talk
3275:talk
3254:talk
3194:talk
3176:talk
3144:talk
3107:talk
3092:talk
3062:talk
3048:talk
2988:talk
2974:talk
2771:talk
2701:talk
2677:talk
2650:talk
2634:talk
2618:talk
2608:and
2591:talk
2566:talk
2547:talk
2533:talk
2513:talk
2497:talk
2473:talk
2458:talk
2413:talk
2391:talk
2370:<
2335:<
2235:<
2198:<
2067:<
2024:<
1868:for
406:<
256:<
3007:sin
2961:0.}
2908:sin
2863:0.5
2790:sin
2733:sin
2719:are
2695:).
2045:if
1831:if
704:if
667:if
597:):
469:lim
460:as
311:m+1
309:∪ A
305:= A
158:...
131:Mid
3354::
3333:)
3315:)
3297:=
3291:⊂
3277:)
3256:)
3237:"
3228:⊆
3222:⊆
3196:)
3178:)
3146:)
3109:)
3094:)
3064:)
3050:)
3010:
2990:)
2976:)
2955:0.
2949:ε
2911:
2903:ε
2869:π
2793:
2773:)
2736:
2703:)
2679:)
2652:)
2636:)
2620:)
2593:)
2568:)
2549:)
2535:)
2515:)
2499:)
2475:)
2460:)
2452:--
2415:)
2393:)
2373:∞
2358:λ
2338:∞
2323:μ
2294:ε
2285:≥
2282:∞
2270:δ
2258:μ
2238:δ
2227:δ
2213:δ
2201:λ
2168:∈
2155:δ
2142:δ
2137:−
2129::=
2124:δ
2103:0}
2091:δ
2070:δ
2053:λ
2027:ε
2010:μ
1994:0}
1982:δ
1965:0}
1950::=
1947:ε
1926:λ
1906:μ
1879:∈
1839:λ
1802:μ
1782:μ
1748:∞
1717:μ
1714:≥
1693:μ
1688:∞
1673:∑
1669:≥
1649:∞
1634:⋃
1625:μ
1619:∞
1611::
1599:μ
1583:0}
1546:λ
1520:∈
1476:∞
1461:∑
1436:λ
1431:∞
1416:∑
1392:∞
1377:⋃
1368:λ
1348:λ
1323:∈
1291:0}
1262:∞
1247:⋃
1238:λ
1196:∞
1181:⋃
1172:μ
1155:∞
1140:∑
1115:μ
1110:∞
1095:∑
1086::
1074:μ
1049:∈
1003:∞
988:⋃
979:λ
976:≤
955:λ
952:≤
906:∞
891:⋃
882:λ
844:∈
834:∈
781:∅
774:μ
754:μ
725:0.
708:λ
695:∞
671:λ
647:↦
633:∞
619:→
606:μ
585:λ
582:⋅
579:∞
576::=
573:μ
529:λ
484:ν
479:∞
476:→
435:⋂
429:ν
409:∞
387:ν
340:ν
259:ε
244:ν
184:.
3329:(
3311:(
3273:(
3252:(
3192:(
3174:(
3155:R
3142:(
3136:R
3132:y
3130:,
3128:x
3126:(
3124:F
3105:(
3090:(
3060:(
3046:(
3030:.
3027:)
3024:x
3020:/
3016:1
3013:(
3004:x
2986:(
2972:(
2928:)
2925:x
2921:/
2917:1
2914:(
2900:+
2897:1
2893:x
2872:)
2866:)
2860:+
2857:n
2854:(
2851:(
2847:/
2843:1
2840:=
2835:n
2831:x
2810:)
2807:x
2803:/
2799:1
2796:(
2787:x
2769:(
2753:)
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2730:x
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2675:(
2648:(
2632:(
2616:(
2589:(
2577:I
2564:(
2545:(
2531:(
2511:(
2495:(
2471:(
2456:(
2446:k
2442:k
2411:(
2389:(
2367:)
2364:A
2361:(
2332:)
2329:A
2326:(
2306:.
2291:=
2288:1
2279:=
2275:)
2266:A
2262:(
2230:2
2222:=
2218:)
2209:A
2205:(
2195:0
2173:B
2164:)
2158:4
2150:,
2145:4
2133:(
2120:A
2097:0
2063:)
2060:A
2057:(
2033:1
2030:=
2020:)
2017:A
2014:(
1988:0
1959:0
1953:1
1884:B
1876:A
1856:0
1853:=
1849:)
1846:A
1843:(
1819:0
1816:=
1812:)
1809:A
1806:(
1751:,
1745:=
1741:)
1734:0
1730:i
1725:A
1721:(
1710:)
1705:n
1701:A
1697:(
1683:1
1680:=
1677:n
1665:)
1659:n
1655:A
1644:1
1641:=
1638:n
1629:(
1622:=
1577:0
1570:)
1563:0
1559:i
1554:A
1550:(
1524:N
1515:0
1511:i
1490:,
1487:0
1484:=
1481:0
1471:1
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1465:n
1457:=
1453:)
1448:n
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1440:(
1426:1
1423:=
1420:n
1412:=
1408:)
1402:n
1398:A
1387:1
1384:=
1381:n
1372:(
1327:N
1320:i
1317:,
1312:i
1308:A
1285:0
1278:)
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1251:n
1242:(
1216:.
1212:)
1206:n
1202:A
1191:1
1188:=
1185:n
1176:(
1169:=
1166:0
1163:=
1160:0
1150:1
1147:=
1144:n
1136:=
1132:)
1127:n
1123:A
1119:(
1105:1
1102:=
1099:n
1053:N
1046:i
1026:0
1023:=
1019:)
1013:n
1009:A
998:1
995:=
992:n
983:(
972:)
967:i
963:A
959:(
949:0
929:0
926:=
922:)
916:n
912:A
901:1
898:=
895:n
886:(
857:N
850:B
838:N
831:n
826:)
821:n
817:A
813:(
791:0
788:=
784:)
778:(
718:)
715:A
712:(
698:,
688:0
685:=
681:)
678:A
675:(
661:,
658:0
652:{
644:A
641:,
637:]
630:,
627:0
623:[
614:B
609::
551:B
512:!
500:)
495:n
491:B
487:(
473:n
448:)
443:n
439:B
432:(
403:)
398:n
394:B
390:(
365:n
361:B
323:m
319:m
315:m
307:m
303:m
298:n
294:n
290:n
253:)
250:A
247:(
234:A
143:.
42::
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