653:
839:
936:
533:
501:
1692:
1809:
1471:
1306:
1132:
1395:
1212:
1054:
171:
278:
349:
2788:
968:
2866:
2883:
1238:
1080:
1901:
740:
853:
2191:
2050:
648:{\displaystyle \lim _{i\rightarrow \infty }\int _{\Omega }\varphi \,\mathrm {d} (c_{i}T_{a,r_{i}\#}\mu )=\int _{\Omega }\varphi \,\mathrm {d} \nu .}
2706:
2537:
428:
2077:
1613:
2698:
730:
2878:
2944:
2484:
2835:
1722:
979:
The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.
2825:
2635:
2308:
1408:
1243:
2873:
2164:
2820:
2714:
2620:
2739:
2719:
2683:
2607:
2327:
2043:
2861:
2640:
2602:
2554:
2766:
1085:
1351:
1168:
1010:
2734:
2724:
2645:
2612:
2243:
2152:
2783:
110:
2688:
2464:
2392:
211:
2773:
2856:
2302:
2233:
1312:
At typical points in the support of a measure, the cone of tangent measures is also closed under translations.
703:
2169:
2625:
2383:
2343:
2036:
33:
289:
2908:
2808:
2630:
2352:
2198:
2544:
2469:
2422:
2417:
2412:
2254:
2137:
2095:
941:
2778:
2744:
2652:
2362:
2317:
2159:
2082:
202:
2761:
2751:
2597:
2561:
2439:
2387:
2116:
2073:
514:
2913:
2673:
2658:
2357:
2238:
2216:
1965:
1930:
510:
1491:
in the circle, the set of tangent measures will just be positive constants times 1-dimensional
2830:
2566:
2527:
2522:
2429:
2347:
2132:
2105:
2006:
1564:
1492:
847:
45:
41:
2847:
2756:
2532:
2517:
2507:
2492:
2459:
2454:
2444:
2322:
2297:
2112:
1996:
1957:
1920:
1912:
1217:
1059:
2018:
1879:
2923:
2903:
2678:
2576:
2549:
2407:
2372:
2292:
2186:
2014:
521:
69:
37:
834:{\displaystyle \limsup _{r\downarrow 0}{\frac {\mu (B(a,2r))}{\mu (B(a,r))}}<\infty }
40:) are a useful tool in geometric measure theory. For example, they are used in proving
2813:
2668:
2663:
2474:
2449:
2402:
2332:
2312:
2272:
2262:
2059:
17:
2938:
2918:
2581:
2502:
2497:
2397:
2367:
2337:
2287:
2282:
2277:
2267:
2181:
2100:
931:{\displaystyle 0<\limsup _{r\downarrow 0}{\frac {\mu (B(a,r))}{r^{s}}}<\infty }
29:
25:
2512:
2434:
2174:
2211:
1985:"Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation"
2377:
61:
2221:
1925:
1855:
Further study of tangent measures and tangent spaces leads to the notion of a
2010:
2203:
2147:
2142:
80:
184:
to a ball of radius 1 centered at 0. With this, we may now zoom in on how
2228:
2087:
1856:
1555: ∈ Ω if — after appropriate rescaling —
1969:
1948:
O'Neil, Toby (1995). "A measure with a large set of tangent measures".
1934:
496:{\displaystyle \lim _{i\rightarrow \infty }c_{i}T_{a,r_{i}\#}\mu =\nu }
2001:
1984:
1638:
1961:
1916:
1687:{\displaystyle \mu _{a,r}{\xrightarrow{*}}\theta H^{k}\lfloor _{P},}
1651:
1531:
There is an associated notion of the tangent space of a measure. A
374:
by looking at what these measures tend to look like in the limit as
2028:
36:. Tangent measures (introduced by David Preiss in his study of
2032:
79:
be an arbitrary point in Ω. We can "zoom in" on a small
1804:{\displaystyle \mu _{a,r}(A)={\frac {1}{r^{n-1}}}\mu (a+rA).}
1498:
In 1995, Toby O'Neil produced an example of a Radon measure
1495:
supported on the line tangent to the circle at that point.
1487:
with uniform measure on that circle. Then, for any point
1466:{\displaystyle T_{x,1\#}\nu \in \mathrm {Tan} (\mu ,a)}
1301:{\displaystyle T_{0,r\#}\nu \in \mathrm {Tan} (\mu ,a)}
722:) is nonempty if one of the following conditions hold:
1882:
1725:
1616:
1411:
1354:
1246:
1220:
1171:
1088:
1062:
1013:
944:
856:
743:
536:
431:
292:
214:
113:
401:
such that there exist sequences of positive numbers
2896:
2844:
2797:
2697:
2590:
2483:
2252:
2125:
2066:
1895:
1803:
1686:
1465:
1389:
1300:
1232:
1206:
1126:
1074:
1048:
962:
930:
833:
647:
495:
370:. We can get information about our measure around
343:
272:
165:
714:. By the weak compactness of Radon measures, Tan(
358:gets smaller, this transformation on the measure
1903:: distribution, rectifiability, and densities".
864:
745:
538:
433:
1876:Preiss, David (1987). "Geometry of measures in
1713:is the translated and rescaled measure given by
1127:{\displaystyle c\nu \in \mathrm {Tan} (\mu ,a)}
1390:{\displaystyle \nu \in \mathrm {Tan} (\mu ,a)}
1207:{\displaystyle \nu \in \mathrm {Tan} (\mu ,a)}
1049:{\displaystyle \nu \in \mathrm {Tan} (\mu ,a)}
2044:
8:
2789:RieszâMarkovâKakutani representation theorem
1672:
335:
308:
166:{\displaystyle T_{a,r}(x)={\frac {x-a}{r}},}
273:{\displaystyle T_{a,r\#}\mu (A)=\mu (a+rA)}
2884:Vitale's random BrunnâMinkowski inequality
2801:
2051:
2037:
2029:
2000:
1924:
1887:
1881:
1763:
1754:
1730:
1724:
1675:
1665:
1639:
1633:
1621:
1615:
1522:) consists of all nonzero Radon measures.
1506:such that, for μ-almost every point
1437:
1416:
1410:
1361:
1353:
1272:
1251:
1245:
1219:
1178:
1170:
1098:
1087:
1061:
1020:
1012:
943:
914:
879:
867:
855:
760:
748:
742:
661:We denote the set of tangent measures of
634:
633:
624:
600:
589:
579:
567:
566:
557:
541:
535:
473:
462:
452:
436:
430:
291:
219:
213:
142:
118:
112:
362:spreads out and enlarges the portion of
32:are used to study the local behavior of
24:are used to study the local behavior of
1868:
410: > 0 and decreasing radii
344:{\displaystyle a+rA=\{a+rx:x\in A\}.}
7:
2897:Applications & related
1145:) of tangent measures of a measure
991:) of tangent measures of a measure
694:) of tangent measures of a measure
1444:
1441:
1438:
1426:
1368:
1365:
1362:
1279:
1276:
1273:
1261:
1185:
1182:
1179:
1105:
1102:
1099:
1027:
1024:
1021:
957:
925:
828:
710:is nonempty on mild conditions on
635:
625:
606:
568:
558:
548:
479:
443:
229:
176:which enlarges the ball of radius
14:
963:{\displaystyle 0<s<\infty }
2826:Lebesgue differentiation theorem
2707:Carathéodory's extension theorem
509:where the limit is taken in the
1989:Interfaces and Free Boundaries
1795:
1780:
1748:
1742:
1643:
1547:-dimensional tangent space of
1460:
1448:
1384:
1372:
1295:
1283:
1201:
1189:
1121:
1109:
1043:
1031:
906:
903:
891:
885:
871:
819:
816:
804:
798:
790:
787:
772:
766:
752:
614:
572:
545:
440:
267:
252:
243:
237:
136:
130:
1:
1483:Suppose we have a circle in
2879:PrĂ©kopaâLeindler inequality
1833:, and the tangent space of
1604: > 0 such that
366:supported around the point
2961:
2821:Lebesgue's density theorem
397:is a second Radon measure
104:), via the transformation
28:, in much the same way as
2945:Measures (measure theory)
2874:MinkowskiâSteiner formula
2804:
2689:Projection-valued measure
1590:dimensional tangent space
1336:) of tangent measures of
56:Consider a Radon measure
2857:Isoperimetric inequality
2836:VitaliâHahnâSaks theorem
2165:Carathéodory's criterion
1983:Röger, Matthias (2004).
846:has positive and finite
419: â 0 such that
34:differentiable manifolds
2862:BrunnâMinkowski theorem
2731:Decomposition theorems
731:asymptotically doubling
2909:Descriptive set theory
2809:Disintegration theorem
2244:Universally measurable
1897:
1805:
1688:
1656:
1535:-dimensional subspace
1467:
1391:
1302:
1234:
1233:{\displaystyle r>0}
1208:
1128:
1076:
1075:{\displaystyle c>0}
1050:
964:
932:
835:
649:
497:
345:
274:
167:
2711:Convergence theorems
2170:Cylindrical Ï-algebra
1898:
1896:{\displaystyle R^{n}}
1806:
1689:
1634:
1468:
1401:is in the support of
1392:
1346:translation invariant
1303:
1235:
1209:
1129:
1077:
1051:
1007:of measures, i.e. if
965:
933:
836:
650:
511:weak-∗ topology
498:
346:
275:
168:
2779:Minkowski inequality
2653:Cylinder set measure
2538:Infinite-dimensional
2153:equivalence relation
2083:Lebesgue integration
1880:
1723:
1614:
1409:
1352:
1244:
1218:
1169:
1086:
1060:
1011:
942:
854:
741:
534:
429:
290:
212:
203:push-forward measure
201:) by looking at the
111:
2774:Hölder's inequality
2636:of random variables
2598:Measurable function
2485:Particular measures
2074:Absolute continuity
1655:
1650:
1574:. More precisely:
1510: ∈
515:continuous function
389:of a Radon measure
42:Marstrand's theorem
2914:Probability theory
2239:Transverse measure
2217:Non-measurable set
2199:Locally measurable
1926:10338.dmlcz/133417
1893:
1801:
1684:
1463:
1387:
1324:in the support of
1298:
1230:
1204:
1163:dilation invariant
1153:in the support of
1124:
1072:
1046:
999:in the support of
960:
928:
878:
831:
759:
645:
552:
493:
447:
341:
270:
163:
2932:
2931:
2892:
2891:
2621:almost everywhere
2567:Spherical measure
2465:Strictly positive
2393:Projection-valued
2133:Almost everywhere
2106:Probability space
1775:
1565:Hausdorff measure
1493:Hausdorff measure
920:
863:
823:
744:
537:
432:
378:approaches zero.
158:
2952:
2867:Milman's reverse
2850:
2848:Lebesgue measure
2802:
2206:
2192:infimum/supremum
2113:Measurable space
2053:
2046:
2039:
2030:
2023:
2022:
2004:
1980:
1974:
1973:
1956:(7): 2217â2220.
1945:
1939:
1938:
1928:
1902:
1900:
1899:
1894:
1892:
1891:
1873:
1810:
1808:
1807:
1802:
1776:
1774:
1773:
1755:
1741:
1740:
1693:
1691:
1690:
1685:
1680:
1679:
1670:
1669:
1657:
1649:
1632:
1631:
1527:Related concepts
1472:
1470:
1469:
1464:
1447:
1430:
1429:
1396:
1394:
1393:
1388:
1371:
1307:
1305:
1304:
1299:
1282:
1265:
1264:
1239:
1237:
1236:
1231:
1213:
1211:
1210:
1205:
1188:
1133:
1131:
1130:
1125:
1108:
1081:
1079:
1078:
1073:
1055:
1053:
1052:
1047:
1030:
969:
967:
966:
961:
937:
935:
934:
929:
921:
919:
918:
909:
880:
877:
840:
838:
837:
832:
824:
822:
793:
761:
758:
654:
652:
651:
646:
638:
629:
628:
610:
609:
605:
604:
584:
583:
571:
562:
561:
551:
513:, i.e., for any
502:
500:
499:
494:
483:
482:
478:
477:
457:
456:
446:
350:
348:
347:
342:
279:
277:
276:
271:
233:
232:
172:
170:
169:
164:
159:
154:
143:
129:
128:
38:rectifiable sets
22:tangent measures
2960:
2959:
2955:
2954:
2953:
2951:
2950:
2949:
2935:
2934:
2933:
2928:
2924:Spectral theory
2904:Convex analysis
2888:
2845:
2840:
2793:
2693:
2641:in distribution
2586:
2479:
2309:Logarithmically
2248:
2204:
2187:Essential range
2121:
2062:
2057:
2027:
2026:
1982:
1981:
1977:
1962:10.2307/2160960
1947:
1946:
1942:
1917:10.2307/1971410
1883:
1878:
1877:
1875:
1874:
1870:
1865:
1846:
1759:
1726:
1721:
1720:
1712:
1671:
1661:
1617:
1612:
1611:
1529:
1480:
1412:
1407:
1406:
1350:
1349:
1328:, the cone Tan(
1247:
1242:
1241:
1216:
1215:
1167:
1166:
1084:
1083:
1058:
1057:
1009:
1008:
977:
940:
939:
910:
881:
852:
851:
794:
762:
739:
738:
684:
620:
596:
585:
575:
553:
532:
531:
522:compact support
469:
458:
448:
427:
426:
418:
409:
387:tangent measure
288:
287:
215:
210:
209:
196:
144:
114:
109:
108:
99:
70:Euclidean space
54:
46:Preiss' theorem
12:
11:
5:
2958:
2956:
2948:
2947:
2937:
2936:
2930:
2929:
2927:
2926:
2921:
2916:
2911:
2906:
2900:
2898:
2894:
2893:
2890:
2889:
2887:
2886:
2881:
2876:
2871:
2870:
2869:
2859:
2853:
2851:
2842:
2841:
2839:
2838:
2833:
2831:Sard's theorem
2828:
2823:
2818:
2817:
2816:
2814:Lifting theory
2805:
2799:
2795:
2794:
2792:
2791:
2786:
2781:
2776:
2771:
2770:
2769:
2767:FubiniâTonelli
2759:
2754:
2749:
2748:
2747:
2742:
2737:
2729:
2728:
2727:
2722:
2717:
2709:
2703:
2701:
2695:
2694:
2692:
2691:
2686:
2681:
2676:
2671:
2666:
2661:
2655:
2650:
2649:
2648:
2646:in probability
2643:
2633:
2628:
2623:
2617:
2616:
2615:
2610:
2605:
2594:
2592:
2588:
2587:
2585:
2584:
2579:
2574:
2569:
2564:
2559:
2558:
2557:
2547:
2542:
2541:
2540:
2530:
2525:
2520:
2515:
2510:
2505:
2500:
2495:
2489:
2487:
2481:
2480:
2478:
2477:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2426:
2425:
2420:
2415:
2405:
2400:
2395:
2390:
2380:
2375:
2370:
2365:
2360:
2355:
2353:Locally finite
2350:
2340:
2335:
2330:
2325:
2320:
2315:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2265:
2259:
2257:
2250:
2249:
2247:
2246:
2241:
2236:
2231:
2226:
2225:
2224:
2214:
2209:
2201:
2196:
2195:
2194:
2184:
2179:
2178:
2177:
2167:
2162:
2157:
2156:
2155:
2145:
2140:
2135:
2129:
2127:
2123:
2122:
2120:
2119:
2110:
2109:
2108:
2098:
2093:
2085:
2080:
2070:
2068:
2067:Basic concepts
2064:
2063:
2060:Measure theory
2058:
2056:
2055:
2048:
2041:
2033:
2025:
2024:
2002:10.4171/IFB/93
1995:(1): 105â133.
1975:
1940:
1911:(3): 537â643.
1890:
1886:
1867:
1866:
1864:
1861:
1853:
1852:
1842:
1821:is called the
1814:
1813:
1812:
1811:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1772:
1769:
1766:
1762:
1758:
1753:
1750:
1747:
1744:
1739:
1736:
1733:
1729:
1715:
1714:
1704:
1697:
1696:
1695:
1694:
1683:
1678:
1674:
1668:
1664:
1660:
1654:
1648:
1645:
1642:
1637:
1630:
1627:
1624:
1620:
1606:
1605:
1600:if there is a
1543:is called the
1528:
1525:
1524:
1523:
1496:
1479:
1476:
1475:
1474:
1462:
1459:
1456:
1453:
1450:
1446:
1443:
1440:
1436:
1433:
1428:
1425:
1422:
1419:
1415:
1386:
1383:
1380:
1377:
1374:
1370:
1367:
1364:
1360:
1357:
1310:
1309:
1297:
1294:
1291:
1288:
1285:
1281:
1278:
1275:
1271:
1268:
1263:
1260:
1257:
1254:
1250:
1229:
1226:
1223:
1203:
1200:
1197:
1194:
1191:
1187:
1184:
1181:
1177:
1174:
1135:
1123:
1120:
1117:
1114:
1111:
1107:
1104:
1101:
1097:
1094:
1091:
1071:
1068:
1065:
1045:
1042:
1039:
1036:
1033:
1029:
1026:
1023:
1019:
1016:
976:
973:
972:
971:
959:
956:
953:
950:
947:
927:
924:
917:
913:
908:
905:
902:
899:
896:
893:
890:
887:
884:
876:
873:
870:
866:
865:lim sup
862:
859:
841:
830:
827:
821:
818:
815:
812:
809:
806:
803:
800:
797:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
757:
754:
751:
747:
746:lim sup
683:
680:
679:
678:
658:
657:
656:
655:
644:
641:
637:
632:
627:
623:
619:
616:
613:
608:
603:
599:
595:
592:
588:
582:
578:
574:
570:
565:
560:
556:
550:
547:
544:
540:
526:
525:
506:
505:
504:
503:
492:
489:
486:
481:
476:
472:
468:
465:
461:
455:
451:
445:
442:
439:
435:
421:
420:
414:
405:
352:
351:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
281:
280:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
231:
228:
225:
222:
218:
192:
174:
173:
162:
157:
153:
150:
147:
141:
138:
135:
132:
127:
124:
121:
117:
95:
60:defined on an
53:
50:
30:tangent spaces
26:Radon measures
18:measure theory
13:
10:
9:
6:
4:
3:
2:
2957:
2946:
2943:
2942:
2940:
2925:
2922:
2920:
2919:Real analysis
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2901:
2899:
2895:
2885:
2882:
2880:
2877:
2875:
2872:
2868:
2865:
2864:
2863:
2860:
2858:
2855:
2854:
2852:
2849:
2843:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2815:
2812:
2811:
2810:
2807:
2806:
2803:
2800:
2798:Other results
2796:
2790:
2787:
2785:
2784:RadonâNikodym
2782:
2780:
2777:
2775:
2772:
2768:
2765:
2764:
2763:
2760:
2758:
2757:Fatou's lemma
2755:
2753:
2750:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2730:
2726:
2723:
2721:
2718:
2716:
2713:
2712:
2710:
2708:
2705:
2704:
2702:
2700:
2696:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2656:
2654:
2651:
2647:
2644:
2642:
2639:
2638:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:Convergence:
2618:
2614:
2611:
2609:
2606:
2604:
2601:
2600:
2599:
2596:
2595:
2593:
2589:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2556:
2553:
2552:
2551:
2548:
2546:
2543:
2539:
2536:
2535:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2490:
2488:
2486:
2482:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2424:
2423:Outer regular
2421:
2419:
2418:Inner regular
2416:
2414:
2413:Borel regular
2411:
2410:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2385:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2345:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2310:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2260:
2258:
2256:
2251:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2223:
2220:
2219:
2218:
2215:
2213:
2210:
2208:
2202:
2200:
2197:
2193:
2190:
2189:
2188:
2185:
2183:
2180:
2176:
2173:
2172:
2171:
2168:
2166:
2163:
2161:
2158:
2154:
2151:
2150:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2130:
2128:
2124:
2118:
2114:
2111:
2107:
2104:
2103:
2102:
2101:Measure space
2099:
2097:
2094:
2092:
2090:
2086:
2084:
2081:
2079:
2075:
2072:
2071:
2069:
2065:
2061:
2054:
2049:
2047:
2042:
2040:
2035:
2034:
2031:
2020:
2016:
2012:
2008:
2003:
1998:
1994:
1990:
1986:
1979:
1976:
1971:
1967:
1963:
1959:
1955:
1951:
1944:
1941:
1936:
1932:
1927:
1922:
1918:
1914:
1910:
1906:
1888:
1884:
1872:
1869:
1862:
1860:
1858:
1850:
1845:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1815:
1798:
1792:
1789:
1786:
1783:
1777:
1770:
1767:
1764:
1760:
1756:
1751:
1745:
1737:
1734:
1731:
1727:
1719:
1718:
1717:
1716:
1711:
1707:
1703:
1699:
1698:
1681:
1676:
1666:
1662:
1658:
1652:
1646:
1640:
1635:
1628:
1625:
1622:
1618:
1610:
1609:
1608:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1580:
1577:
1576:
1575:
1573:
1569:
1566:
1563:-dimensional
1562:
1559:"looks like"
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1526:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1494:
1490:
1486:
1482:
1481:
1477:
1457:
1454:
1451:
1434:
1431:
1423:
1420:
1417:
1413:
1404:
1400:
1381:
1378:
1375:
1358:
1355:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1320:almost every
1319:
1315:
1314:
1313:
1292:
1289:
1286:
1269:
1266:
1258:
1255:
1252:
1248:
1227:
1224:
1221:
1198:
1195:
1192:
1175:
1172:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1137:The cone Tan(
1136:
1118:
1115:
1112:
1095:
1092:
1089:
1069:
1066:
1063:
1040:
1037:
1034:
1017:
1014:
1006:
1002:
998:
994:
990:
986:
982:
981:
980:
974:
954:
951:
948:
945:
922:
915:
911:
900:
897:
894:
888:
882:
874:
868:
860:
857:
849:
848:upper density
845:
842:
825:
813:
810:
807:
801:
795:
784:
781:
778:
775:
769:
763:
755:
749:
736:
732:
728:
725:
724:
723:
721:
717:
713:
709:
705:
701:
697:
693:
689:
681:
676:
672:
668:
664:
660:
659:
642:
639:
630:
621:
617:
611:
601:
597:
593:
590:
586:
580:
576:
563:
554:
542:
530:
529:
528:
527:
523:
519:
516:
512:
508:
507:
490:
487:
484:
474:
470:
466:
463:
459:
453:
449:
437:
425:
424:
423:
422:
417:
413:
408:
404:
400:
396:
393:at the point
392:
388:
384:
381:
380:
379:
377:
373:
369:
365:
361:
357:
338:
332:
329:
326:
323:
320:
317:
314:
311:
305:
302:
299:
296:
293:
286:
285:
284:
264:
261:
258:
255:
249:
246:
240:
234:
226:
223:
220:
216:
208:
207:
206:
204:
200:
195:
191:
187:
183:
179:
160:
155:
151:
148:
145:
139:
133:
125:
122:
119:
115:
107:
106:
105:
103:
98:
94:
90:
86:
82:
78:
74:
71:
68:-dimensional
67:
63:
59:
51:
49:
47:
43:
39:
35:
31:
27:
23:
19:
2699:Main results
2571:
2435:Set function
2363:Metric outer
2318:Decomposable
2175:Cylinder set
2088:
1992:
1988:
1978:
1953:
1949:
1943:
1908:
1904:
1871:
1854:
1848:
1843:
1841:is denoted T
1838:
1834:
1830:
1826:
1823:multiplicity
1822:
1818:
1709:
1705:
1701:
1601:
1597:
1593:
1589:
1585:
1581:
1578:
1571:
1567:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1530:
1519:
1515:
1511:
1507:
1503:
1499:
1488:
1484:
1402:
1398:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1311:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1004:
1000:
996:
992:
988:
984:
983:The set Tan(
978:
843:
734:
726:
719:
715:
711:
707:
699:
695:
691:
687:
686:The set Tan(
685:
674:
670:
666:
662:
517:
415:
411:
406:
402:
398:
394:
390:
386:
382:
375:
371:
367:
363:
359:
355:
353:
282:
198:
193:
189:
185:
181:
177:
175:
101:
96:
92:
88:
84:
76:
72:
65:
57:
55:
21:
15:
2659:compact set
2626:of measures
2562:Pushforward
2555:Projections
2545:Logarithmic
2388:Probability
2378:Pre-measure
2160:Borel space
2078:of measures
1817:The number
1579:Definition.
1149:at a point
995:at a point
698:at a point
383:Definition.
205:defined by
188:behaves on
62:open subset
2631:in measure
2358:Maximising
2328:Equivalent
2222:Vitali set
1863:References
1348:, i.e. if
1165:, i.e. if
975:Properties
524:in Ω,
83:of radius
64:Ω of
52:Definition
2745:Maharam's
2715:Dominated
2528:Intensity
2523:Hausdorff
2430:Saturated
2348:Invariant
2253:Types of
2212:Ï-algebra
2182:đ-system
2148:Borel set
2143:Baire set
2011:1463-9963
1950:Proc. AMS
1905:Ann. Math
1778:μ
1768:−
1728:μ
1673:⌊
1659:θ
1653:∗
1644:→
1619:μ
1452:μ
1435:∈
1432:ν
1427:#
1376:μ
1359:∈
1356:ν
1287:μ
1270:∈
1267:ν
1262:#
1193:μ
1176:∈
1173:ν
1113:μ
1096:∈
1093:ν
1035:μ
1018:∈
1015:ν
958:∞
938:for some
926:∞
883:μ
872:↓
829:∞
796:μ
764:μ
753:↓
682:Existence
640:ν
631:φ
626:Ω
622:∫
612:μ
607:#
564:φ
559:Ω
555:∫
549:∞
546:→
491:ν
485:μ
480:#
444:∞
441:→
330:∈
250:μ
235:μ
230:#
149:−
81:open ball
2939:Category
2762:Fubini's
2752:Egorov's
2720:Monotone
2679:variable
2657:Random:
2608:Strongly
2533:Lebesgue
2518:Harmonic
2508:Gaussian
2493:Counting
2460:Spectral
2455:Singular
2445:s-finite
2440:Ï-finite
2323:Discrete
2298:Complete
2255:Measures
2229:Null set
2117:function
1857:varifold
1636:→
1478:Examples
75:and let
2674:process
2669:measure
2664:element
2603:Bochner
2577:Trivial
2572:Tangent
2550:Product
2408:Regular
2386:)
2373:Perfect
2346:)
2311:)
2303:Content
2293:Complex
2234:Support
2207:-system
2096:Measure
2019:2047075
1970:2160960
1935:1971410
1584:is the
1518:,
1405:, then
1332:,
1240:, then
1141:,
1082:, then
987:,
850:, i.e.
737:, i.e.
718:,
704:support
702:in the
690:,
673:,
669:by Tan(
87:around
2740:Jordan
2725:Vitali
2684:vector
2613:Weakly
2475:Vector
2450:Signed
2403:Random
2344:Quasi-
2333:Finite
2313:Convex
2273:Banach
2263:Atomic
2091:spaces
2076:
2017:
2009:
1968:
1933:
1849:μ
1835:μ
1827:μ
1819:θ
1702:μ
1700:where
1602:θ
1594:μ
1557:μ
1549:μ
1516:μ
1514:, Tan(
1500:μ
1403:ν
1338:μ
1330:μ
1326:μ
1318:μ
1159:d-cone
1155:μ
1147:μ
1139:μ
1001:μ
993:μ
985:μ
844:μ
727:μ
716:μ
712:μ
708:μ
696:μ
688:μ
671:μ
663:μ
518:φ
399:ν
391:μ
364:μ
360:μ
283:where
186:μ
180:about
58:μ
2582:Young
2503:Euler
2498:Dirac
2470:Tight
2398:Radon
2368:Outer
2338:Inner
2288:Brown
2283:Borel
2278:Besov
2268:Baire
1966:JSTOR
1931:JSTOR
1157:is a
1003:is a
520:with
2846:For
2735:Hahn
2591:Maps
2513:Haar
2384:Sub-
2138:Atom
2126:Sets
2007:ISSN
1397:and
1225:>
1214:and
1067:>
1056:and
1005:cone
955:<
949:<
923:<
861:<
826:<
44:and
1997:doi
1958:doi
1954:123
1921:hdl
1913:doi
1909:125
1837:at
1829:at
1825:of
1596:at
1592:of
1570:on
1551:at
1539:of
1502:on
1344:is
1340:at
1316:At
1161:or
733:at
729:is
706:of
665:at
539:lim
434:lim
354:As
16:In
2941::
2015:MR
2013:.
2005:.
1991:.
1987:.
1964:.
1952:.
1929:.
1919:.
1907:.
1859:.
1851:).
677:).
385:A
91:,
48:.
20:,
2382:(
2342:(
2307:(
2205:Ï
2115:/
2089:L
2052:e
2045:t
2038:v
2021:.
1999::
1993:6
1972:.
1960::
1937:.
1923::
1915::
1889:n
1885:R
1847:(
1844:a
1839:a
1831:a
1799:.
1796:)
1793:A
1790:r
1787:+
1784:a
1781:(
1771:1
1765:n
1761:r
1757:1
1752:=
1749:)
1746:A
1743:(
1738:r
1735:,
1732:a
1710:r
1708:,
1706:a
1682:,
1677:P
1667:k
1663:H
1647:0
1641:r
1629:r
1626:,
1623:a
1598:a
1588:-
1586:k
1582:P
1572:P
1568:H
1561:k
1553:a
1545:k
1541:R
1537:P
1533:k
1520:a
1512:R
1508:a
1504:R
1489:a
1485:R
1473:.
1461:)
1458:a
1455:,
1449:(
1445:n
1442:a
1439:T
1424:1
1421:,
1418:x
1414:T
1399:x
1385:)
1382:a
1379:,
1373:(
1369:n
1366:a
1363:T
1342:a
1334:a
1322:a
1308:.
1296:)
1293:a
1290:,
1284:(
1280:n
1277:a
1274:T
1259:r
1256:,
1253:0
1249:T
1228:0
1222:r
1202:)
1199:a
1196:,
1190:(
1186:n
1183:a
1180:T
1151:a
1143:a
1134:.
1122:)
1119:a
1116:,
1110:(
1106:n
1103:a
1100:T
1090:c
1070:0
1064:c
1044:)
1041:a
1038:,
1032:(
1028:n
1025:a
1022:T
997:a
989:a
970:.
952:s
946:0
916:s
912:r
907:)
904:)
901:r
898:,
895:a
892:(
889:B
886:(
875:0
869:r
858:0
820:)
817:)
814:r
811:,
808:a
805:(
802:B
799:(
791:)
788:)
785:r
782:2
779:,
776:a
773:(
770:B
767:(
756:0
750:r
735:a
720:a
700:a
692:a
675:a
667:a
643:.
636:d
618:=
615:)
602:i
598:r
594:,
591:a
587:T
581:i
577:c
573:(
569:d
543:i
488:=
475:i
471:r
467:,
464:a
460:T
454:i
450:c
438:i
416:i
412:r
407:i
403:c
395:a
376:r
372:a
368:a
356:r
339:.
336:}
333:A
327:x
324::
321:x
318:r
315:+
312:a
309:{
306:=
303:A
300:r
297:+
294:a
268:)
265:A
262:r
259:+
256:a
253:(
247:=
244:)
241:A
238:(
227:r
224:,
221:a
217:T
199:a
197:(
194:r
190:B
182:a
178:r
161:,
156:r
152:a
146:x
140:=
137:)
134:x
131:(
126:r
123:,
120:a
116:T
102:a
100:(
97:r
93:B
89:a
85:r
77:a
73:R
66:n
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