2878:
2697:
1915:
1001:
1089:
190:
111:
1832:
1767:
925:
2498:
1703:
356:
474:
584:
1449:
1329:
476:
has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.
2873:{\displaystyle \{\left\{\left(e^{a},\ e^{a}\right):a\in R\right\},\times \}{\text{ on }}L\quad {\text{ and }}\quad \{\left\{\left(e^{a},\ e^{-a}\right):a\in R\right\},\times \}{\text{ on }}H.}
2123:
2028:
2275:
1127:
2361:
2162:
2912:
1481:
736:
652:
513:
424:
390:
1623:
2616:
1844:
306:
277:
248:
219:
2554:
1529:
3026:
2688:
1561:
777:
613:
2978:
1653:
2431:
2408:
2644:
1377:
938:
839:
1202:
2234:
810:
1010:
1587:. In the study of physical magnitudes, the order of decades provides positive and negative ordinals referring to an ordinal scale implicit in the ratio scale.
1274:
1248:
1225:
698:
2936:
1581:
1397:
3901:
3099:
124:
3979:
45:
3996:
1772:
1712:
850:
2444:
3304:
3163:
1658:
311:
429:
2277:
just as the
Lebesgue measure is invariant under addition. In the context of topological groups, this measure is an example of a
533:
3819:
3650:
1406:
1286:
3190:
3811:
3597:
3991:
1204:
which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For rational
2037:
1983:
3948:
3938:
2983:
The realms of business and science abound in ratios, and any change in ratios draws attention. The study refers to
3748:
3421:
2239:
1094:
3277:
3986:
3933:
3827:
3733:
1969:
3852:
3832:
3796:
3720:
3440:
3156:
2334:
2301:
2132:
2885:
1454:
706:
625:
486:
395:
361:
4057:
3974:
3753:
3715:
3667:
1910:{\displaystyle \operatorname {SL} (n,\mathbb {R} )\triangleleft \operatorname {GL} ^{+}(n,\mathbb {R} ),}
1277:
587:
17:
1597:
3879:
3847:
3837:
3758:
3725:
3356:
3265:
2559:
2500:
the first quadrant of the
Cartesian plane. The quadrant itself is divided into four parts by the line
282:
253:
224:
195:
3896:
3801:
3577:
3505:
2503:
1498:
3886:
3969:
3415:
3346:
3001:
2939:
2649:
1534:
3282:
4062:
3738:
3496:
3456:
3149:
741:
592:
996:{\displaystyle \operatorname {floor} :[1,\infty )\to \mathbb {N} ,\,x\mapsto \lfloor x\rfloor ,}
4021:
3921:
3743:
3465:
3311:
2984:
2957:
2031:
1938:
1632:
928:
2413:
2390:
3582:
3535:
3530:
3525:
3367:
3250:
3208:
3121:
2623:
2325:
2126:
1934:
1352:
1347:
815:
671:
655:
1135:
1084:{\displaystyle \operatorname {excess} :[1,\infty )\to (0,1),\,x\mapsto x-\lfloor x\rfloor ,}
3891:
3857:
3765:
3475:
3430:
3272:
3195:
2372:
2317:
1958:
1918:
1706:
1488:
1251:
2183:
786:
524:
8:
3874:
3864:
3710:
3674:
3500:
3229:
3186:
2915:
2691:
2313:
1954:
1492:
780:
3552:
1256:
1230:
1207:
680:
4026:
3786:
3771:
3470:
3351:
3329:
3119:
Kist, Joseph; Leetsma, Sanford (1970). "Additive semigroups of positive real numbers".
3095:
3051:
2921:
2321:
1584:
1566:
1382:
1130:
1972:
of
Eudoxus was developed, which is equivalent to a theory of positive real numbers."
3943:
3679:
3640:
3635:
3542:
3460:
3245:
3218:
2368:
2293:
2285:
2177:
2173:
1965:
1004:
667:
35:
3960:
3869:
3645:
3630:
3620:
3605:
3572:
3567:
3557:
3435:
3410:
3225:
3130:
3103:
3033:
2376:
2169:
1950:
1340:
185:{\displaystyle \mathbb {R} _{\geq 0}=\left\{x\in \mathbb {R} \mid x\geq 0\right\},}
4036:
4016:
3791:
3689:
3684:
3662:
3520:
3485:
3405:
3299:
1839:
1591:
1344:
106:{\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},}
3072:
3926:
3781:
3776:
3587:
3562:
3515:
3445:
3425:
3385:
3375:
3172:
2996:
1835:
1827:{\displaystyle \operatorname {GL} ^{+}(n,\mathbb {R} )\to \mathbb {R} _{>0}}
932:
528:
520:
4051:
4031:
3694:
3615:
3610:
3510:
3480:
3450:
3400:
3395:
3390:
3380:
3294:
3213:
842:
480:
3625:
3547:
3287:
2384:
2278:
3324:
3490:
2380:
2297:
1946:
1626:
1332:
114:
31:
1762:{\displaystyle \mathrm {GL} (n,\mathbb {R} )\to \mathbb {R} ^{\times }.}
3334:
3134:
3316:
3260:
3255:
3045:
1942:
1922:
1583:
is the integer in the doubly infinite progression, and is called the
663:
1964:
An early expression of ratio scale was articulated geometrically by
920:{\displaystyle \mathbb {R} _{>0}=(0,1)\cup \{1\}\cup (1,\infty )}
3341:
3200:
2364:
2165:
701:
659:
2493:{\displaystyle Q=\mathbb {R} _{>0}\times \mathbb {R} _{>0},}
1949:
are equal. Other ratios are compared to one by logarithms, often
1769:
Restricting to matrices with a positive determinant gives the map
3141:
2289:
2172:
on the real numbers under the logarithm: it is the length on the
1968:: "it was ... in geometrical language that the general theory of
1400:
2284:
The utility of this measure is shown in its use for describing
1227:
the sequence terminates with an exact fractional expression of
1698:{\displaystyle \mathrm {M} (n,\mathbb {R} )\to \mathbb {R} .}
658:
under addition, multiplication, and division. It inherits a
351:{\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}}
469:{\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}}
579:{\displaystyle z=|z|\mathrm {e} ^{\mathrm {i} \varphi },}
2690:
is the central point. It is the identity element of two
1705:
Restricting to invertible matrices gives a map from the
738:
of its integral powers has three different fates: When
250:
are ambiguously used for either of these, the notation
1444:{\displaystyle \left(\mathbb {R} _{>0},>\right)}
1324:{\displaystyle \left(\mathbb {R} _{>0},>\right)}
3054: â Number property of being positive or negative
3004:
2960:
2924:
2888:
2700:
2652:
2626:
2562:
2506:
2447:
2416:
2393:
2337:
2242:
2186:
2135:
2040:
1986:
1957:
used in science and technology, expressed in various
1847:
1775:
1715:
1661:
1635:
1600:
1569:
1537:
1501:
1457:
1409:
1385:
1355:
1289:
1259:
1233:
1210:
1138:
1097:
1013:
941:
853:
818:
789:
744:
709:
683:
628:
595:
536:
489:
432:
398:
364:
314:
285:
256:
227:
198:
127:
48:
2375:; taking logarithms (with a choice of base giving a
3020:
2972:
2930:
2906:
2872:
2682:
2638:
2610:
2548:
2492:
2425:
2402:
2355:
2300:, the dimensionless quantities are referred to as
2269:
2228:
2156:
2117:
2022:
1909:
1826:
1761:
1697:
1647:
1617:
1575:
1555:
1523:
1475:
1443:
1391:
1371:
1323:
1268:
1242:
1219:
1196:
1121:
1083:
995:
919:
833:
804:
771:
730:
692:
646:
607:
578:
507:
468:
418:
384:
350:
300:
271:
242:
213:
184:
105:
666:and, thus, has the structure of a multiplicative
27:Subset of real numbers that are greater than zero
4049:
2410:), and its units (the finite numbers, excluding
1953:using base 10. The ratio scale then segments by
1937:the ratio scale provides the finest detail. The
931:function exchanges the intervals. The functions
2980:is a resolution of the types of group action.
2118:{\displaystyle \mu ()=\log(b/a)=\log b-\log a}
2023:{\displaystyle \subseteq \mathbb {R} _{>0}}
3157:
1655:matrices over the reals to the real numbers:
3902:RieszâMarkovâKakutani representation theorem
2856:
2786:
2768:
2701:
2602:
2569:
2543:
2513:
1075:
1069:
987:
981:
896:
890:
3118:
2433:) correspond to the positive real numbers.
3997:Vitale's random BrunnâMinkowski inequality
3164:
3150:
2296:. For purposes of international standards
2270:{\displaystyle z\in \mathbb {R} _{>0},}
2954:profile the activity in the product, and
2891:
2474:
2456:
2340:
2251:
2138:
2007:
1897:
1864:
1811:
1799:
1746:
1734:
1688:
1677:
1608:
1460:
1417:
1297:
1122:{\displaystyle x\in \mathbb {R} _{>0}}
1106:
1056:
974:
967:
856:
631:
492:
445:
401:
367:
327:
288:
259:
230:
201:
158:
130:
79:
51:
3657:
527:. The real positive axis corresponds to
192:also include zero. Although the symbols
523:. This ray is used as reference in the
519:, and is usually drawn as a horizontal
14:
4050:
1975:
1451:and serves to section it for access.
1091:have been used to describe an element
3145:
2356:{\displaystyle \mathbb {R} _{\geq 0}}
2157:{\displaystyle \mathbb {R} _{>0},}
1921:, expresses the positive reals as a
4010:Applications & related
2907:{\displaystyle \mathbb {R} _{>0}}
2312:The non-negative reals serve as the
1941:function takes a value of one when
1476:{\displaystyle \mathbb {R} _{>0}}
731:{\displaystyle \left\{x^{n}\right\}}
647:{\displaystyle \mathbb {R} _{>0}}
508:{\displaystyle \mathbb {R} _{>0}}
419:{\displaystyle \mathbb {R} _{*}^{+}}
385:{\displaystyle \mathbb {R} _{+}^{*}}
812:the sequence is constant; and when
24:
3171:
3100:Mechanization of the World-Picture
2420:
2397:
2292:, among other applications of the
1720:
1717:
1663:
1618:{\displaystyle n\in \mathbb {N} ,}
1029:
957:
911:
564:
558:
25:
4074:
2611:{\displaystyle H=\{(x,y):xy=1\}.}
677:For a given positive real number
3939:Lebesgue differentiation theorem
3820:Carathéodory's extension theorem
301:{\displaystyle \mathbb {R} ^{+}}
272:{\displaystyle \mathbb {R} _{+}}
243:{\displaystyle \mathbb {R} ^{+}}
214:{\displaystyle \mathbb {R} _{+}}
117:that are greater than zero. The
3112:
2995:axis indicates a change in the
2785:
2779:
2549:{\displaystyle L=\{(x,y):x=y\}}
2307:
2180:with respect to multiplication
1524:{\displaystyle a\times 10^{b},}
3089:
3065:
2942:. The one-parameter subgroups
2677:
2665:
2584:
2572:
2528:
2516:
2223:
2205:
2202:
2199:
2187:
2088:
2074:
2062:
2059:
2047:
2044:
1999:
1987:
1928:
1901:
1887:
1868:
1854:
1834:; interpreting the image as a
1806:
1803:
1789:
1741:
1738:
1724:
1684:
1681:
1667:
1060:
1050:
1038:
1035:
1032:
1020:
978:
963:
960:
948:
914:
902:
884:
872:
763:
751:
552:
544:
525:polar form of a complex number
13:
1:
3058:
3021:{\displaystyle {\sqrt {xy}},}
2683:{\displaystyle L\cap H=(1,1)}
1556:{\displaystyle 1\leq a<10}
1491:. Elements may be written in
617:
7:
3992:PrĂ©kopaâLeindler inequality
3048: â Algebraic structure
3039:
2556:and the standard hyperbola
1278:periodic continued fraction
772:{\displaystyle x\in (0,1),}
608:{\displaystyle \varphi =0.}
10:
4079:
3934:Lebesgue's density theorem
1709:to non-zero real numbers:
4009:
3987:MinkowskiâSteiner formula
3957:
3917:
3910:
3810:
3802:Projection-valued measure
3703:
3596:
3365:
3238:
3179:
3073:"positive number in nLab"
2973:{\displaystyle L\times H}
2436:
2387:(with 0 corresponding to
1648:{\displaystyle n\times n}
119:non-negative real numbers
3970:Isoperimetric inequality
3949:VitaliâHahnâSaks theorem
3278:Carathéodory's criterion
2940:direct product of groups
2426:{\displaystyle -\infty }
2403:{\displaystyle -\infty }
3975:BrunnâMinkowski theorem
3844:Decomposition theorems
2639:{\displaystyle L\cup H}
2367:structure (0 being the
1372:{\displaystyle 10^{n},}
1276:the sequence becomes a
834:{\displaystyle x>1,}
515:is identified with the
113:is the subset of those
4022:Descriptive set theory
3922:Disintegration theorem
3357:Universally measurable
3022:
2985:hyperbolic coordinates
2974:
2932:
2908:
2874:
2694:that intersect there:
2684:
2646:forms a trident while
2640:
2612:
2550:
2494:
2427:
2404:
2357:
2271:
2230:
2158:
2129:on certain subsets of
2119:
2024:
1911:
1828:
1763:
1699:
1649:
1619:
1577:
1557:
1525:
1477:
1445:
1393:
1373:
1325:
1270:
1244:
1221:
1198:
1197:{\displaystyle \left,}
1123:
1085:
997:
929:multiplicative inverse
921:
835:
806:
773:
732:
694:
648:
609:
580:
509:
470:
420:
386:
352:
302:
273:
244:
215:
186:
107:
3824:Convergence theorems
3283:Cylindrical Ï-algebra
3122:Mathematische Annalen
3028:while a change along
3023:
2991:. Motion against the
2975:
2933:
2909:
2875:
2685:
2641:
2613:
2551:
2495:
2428:
2405:
2358:
2331:Including 0, the set
2272:
2231:
2164:corresponding to the
2159:
2120:
2025:
1935:levels of measurement
1912:
1829:
1764:
1700:
1650:
1620:
1578:
1558:
1526:
1478:
1446:
1394:
1374:
1348:geometric progression
1326:
1271:
1245:
1222:
1199:
1124:
1086:
998:
922:
836:
807:
774:
733:
695:
672:topological semigroup
649:
610:
581:
510:
471:
421:
387:
353:
303:
274:
245:
216:
187:
108:
40:positive real numbers
3892:Minkowski inequality
3766:Cylinder set measure
3651:Infinite-dimensional
3266:equivalence relation
3196:Lebesgue integration
3002:
2958:
2922:
2886:
2698:
2692:one-parameter groups
2650:
2624:
2560:
2504:
2445:
2414:
2391:
2373:probability semiring
2335:
2288:and noise levels in
2240:
2229:{\displaystyle \to }
2184:
2176:. In fact, it is an
2133:
2038:
1984:
1959:units of measurement
1919:special linear group
1845:
1773:
1713:
1707:general linear group
1659:
1633:
1598:
1567:
1535:
1499:
1489:level of measurement
1455:
1407:
1383:
1353:
1287:
1257:
1252:quadratic irrational
1231:
1208:
1136:
1095:
1011:
939:
851:
816:
805:{\displaystyle x=1,}
787:
742:
707:
681:
626:
593:
534:
487:
430:
396:
362:
312:
283:
254:
225:
196:
125:
46:
3887:Hölder's inequality
3749:of random variables
3711:Measurable function
3598:Particular measures
3187:Absolute continuity
1976:Logarithmic measure
1955:orders of magnitude
1493:scientific notation
1403:, lies entirely in
415:
381:
4058:Topological groups
4027:Probability theory
3352:Transverse measure
3330:Non-measurable set
3312:Locally measurable
3135:10.1007/BF01350237
3096:E. J. Dijksterhuis
3052:Sign (mathematics)
3018:
2970:
2928:
2904:
2870:
2680:
2636:
2608:
2546:
2490:
2423:
2400:
2353:
2286:stellar magnitudes
2267:
2226:
2154:
2115:
2020:
1907:
1824:
1759:
1695:
1645:
1615:
1573:
1553:
1521:
1473:
1441:
1389:
1369:
1321:
1269:{\displaystyle x,}
1266:
1243:{\displaystyle x,}
1240:
1220:{\displaystyle x,}
1217:
1194:
1131:continued fraction
1119:
1081:
993:
917:
831:
802:
769:
728:
693:{\displaystyle x,}
690:
670:or of an additive
644:
605:
576:
517:positive real axis
505:
466:
416:
399:
382:
365:
348:
298:
269:
240:
211:
182:
103:
4045:
4044:
4005:
4004:
3734:almost everywhere
3680:Spherical measure
3578:Strictly positive
3506:Projection-valued
3246:Almost everywhere
3219:Probability space
3013:
2931:{\displaystyle Q}
2862:
2814:
2783:
2774:
2729:
2369:additive identity
2294:logarithmic scale
2178:invariant measure
2174:logarithmic scale
1629:gives a map from
1576:{\displaystyle b}
1392:{\displaystyle n}
668:topological group
16:(Redirected from
4070:
3980:Milman's reverse
3963:
3961:Lebesgue measure
3915:
3914:
3319:
3305:infimum/supremum
3226:Measurable space
3166:
3159:
3152:
3143:
3142:
3138:
3106:
3104:Internet Archive
3093:
3087:
3086:
3084:
3083:
3069:
3034:hyperbolic angle
3032:indicates a new
3027:
3025:
3024:
3019:
3014:
3006:
2979:
2977:
2976:
2971:
2937:
2935:
2934:
2929:
2913:
2911:
2910:
2905:
2903:
2902:
2894:
2879:
2877:
2876:
2871:
2863:
2860:
2849:
2845:
2832:
2828:
2827:
2826:
2812:
2808:
2807:
2784:
2781:
2775:
2772:
2761:
2757:
2744:
2740:
2739:
2738:
2727:
2723:
2722:
2689:
2687:
2686:
2681:
2645:
2643:
2642:
2637:
2617:
2615:
2614:
2609:
2555:
2553:
2552:
2547:
2499:
2497:
2496:
2491:
2486:
2485:
2477:
2468:
2467:
2459:
2432:
2430:
2429:
2424:
2409:
2407:
2406:
2401:
2377:logarithmic unit
2371:), known as the
2362:
2360:
2359:
2354:
2352:
2351:
2343:
2328:in mathematics.
2276:
2274:
2273:
2268:
2263:
2262:
2254:
2235:
2233:
2232:
2227:
2170:Lebesgue measure
2163:
2161:
2160:
2155:
2150:
2149:
2141:
2124:
2122:
2121:
2116:
2084:
2029:
2027:
2026:
2021:
2019:
2018:
2010:
1951:common logarithm
1916:
1914:
1913:
1908:
1900:
1883:
1882:
1867:
1833:
1831:
1830:
1825:
1823:
1822:
1814:
1802:
1785:
1784:
1768:
1766:
1765:
1760:
1755:
1754:
1749:
1737:
1723:
1704:
1702:
1701:
1696:
1691:
1680:
1666:
1654:
1652:
1651:
1646:
1624:
1622:
1621:
1616:
1611:
1592:classical groups
1590:In the study of
1582:
1580:
1579:
1574:
1562:
1560:
1559:
1554:
1530:
1528:
1527:
1522:
1517:
1516:
1482:
1480:
1479:
1474:
1472:
1471:
1463:
1450:
1448:
1447:
1442:
1440:
1436:
1429:
1428:
1420:
1398:
1396:
1395:
1390:
1378:
1376:
1375:
1370:
1365:
1364:
1341:well-ordered set
1330:
1328:
1327:
1322:
1320:
1316:
1309:
1308:
1300:
1283:The ordered set
1275:
1273:
1272:
1267:
1249:
1247:
1246:
1241:
1226:
1224:
1223:
1218:
1203:
1201:
1200:
1195:
1190:
1186:
1179:
1178:
1166:
1165:
1153:
1152:
1128:
1126:
1125:
1120:
1118:
1117:
1109:
1090:
1088:
1087:
1082:
1002:
1000:
999:
994:
970:
926:
924:
923:
918:
868:
867:
859:
841:the sequence is
840:
838:
837:
832:
811:
809:
808:
803:
778:
776:
775:
770:
737:
735:
734:
729:
727:
723:
722:
699:
697:
696:
691:
653:
651:
650:
645:
643:
642:
634:
614:
612:
611:
606:
585:
583:
582:
577:
572:
571:
567:
561:
555:
547:
514:
512:
511:
506:
504:
503:
495:
475:
473:
472:
467:
465:
461:
448:
425:
423:
422:
417:
414:
409:
404:
391:
389:
388:
383:
380:
375:
370:
357:
355:
354:
349:
347:
343:
330:
307:
305:
304:
299:
297:
296:
291:
278:
276:
275:
270:
268:
267:
262:
249:
247:
246:
241:
239:
238:
233:
220:
218:
217:
212:
210:
209:
204:
191:
189:
188:
183:
178:
174:
161:
142:
141:
133:
112:
110:
109:
104:
99:
95:
82:
63:
62:
54:
21:
4078:
4077:
4073:
4072:
4071:
4069:
4068:
4067:
4048:
4047:
4046:
4041:
4037:Spectral theory
4017:Convex analysis
4001:
3958:
3953:
3906:
3806:
3754:in distribution
3699:
3592:
3422:Logarithmically
3361:
3317:
3300:Essential range
3234:
3175:
3170:
3115:
3110:
3109:
3102:, page 51, via
3094:
3090:
3081:
3079:
3071:
3070:
3066:
3061:
3042:
3005:
3003:
3000:
2999:
2959:
2956:
2955:
2923:
2920:
2919:
2895:
2890:
2889:
2887:
2884:
2883:
2859:
2819:
2815:
2803:
2799:
2798:
2794:
2793:
2789:
2782: and
2780:
2771:
2734:
2730:
2718:
2714:
2713:
2709:
2708:
2704:
2699:
2696:
2695:
2651:
2648:
2647:
2625:
2622:
2621:
2561:
2558:
2557:
2505:
2502:
2501:
2478:
2473:
2472:
2460:
2455:
2454:
2446:
2443:
2442:
2439:
2415:
2412:
2411:
2392:
2389:
2388:
2344:
2339:
2338:
2336:
2333:
2332:
2310:
2255:
2250:
2249:
2241:
2238:
2237:
2185:
2182:
2181:
2142:
2137:
2136:
2134:
2131:
2130:
2080:
2039:
2036:
2035:
2011:
2006:
2005:
1985:
1982:
1981:
1978:
1931:
1896:
1878:
1874:
1863:
1846:
1843:
1842:
1840:normal subgroup
1815:
1810:
1809:
1798:
1780:
1776:
1774:
1771:
1770:
1750:
1745:
1744:
1733:
1716:
1714:
1711:
1710:
1687:
1676:
1662:
1660:
1657:
1656:
1634:
1631:
1630:
1607:
1599:
1596:
1595:
1568:
1565:
1564:
1536:
1533:
1532:
1512:
1508:
1500:
1497:
1496:
1464:
1459:
1458:
1456:
1453:
1452:
1421:
1416:
1415:
1414:
1410:
1408:
1405:
1404:
1384:
1381:
1380:
1360:
1356:
1354:
1351:
1350:
1345:doubly infinite
1301:
1296:
1295:
1294:
1290:
1288:
1285:
1284:
1258:
1255:
1254:
1232:
1229:
1228:
1209:
1206:
1205:
1174:
1170:
1161:
1157:
1148:
1144:
1143:
1139:
1137:
1134:
1133:
1110:
1105:
1104:
1096:
1093:
1092:
1012:
1009:
1008:
966:
940:
937:
936:
860:
855:
854:
852:
849:
848:
817:
814:
813:
788:
785:
784:
743:
740:
739:
718:
714:
710:
708:
705:
704:
682:
679:
678:
635:
630:
629:
627:
624:
623:
620:
594:
591:
590:
563:
562:
557:
556:
551:
543:
535:
532:
531:
529:complex numbers
496:
491:
490:
488:
485:
484:
444:
437:
433:
431:
428:
427:
410:
405:
400:
397:
394:
393:
376:
371:
366:
363:
360:
359:
326:
319:
315:
313:
310:
309:
292:
287:
286:
284:
281:
280:
263:
258:
257:
255:
252:
251:
234:
229:
228:
226:
223:
222:
205:
200:
199:
197:
194:
193:
157:
150:
146:
134:
129:
128:
126:
123:
122:
78:
71:
67:
55:
50:
49:
47:
44:
43:
28:
23:
22:
15:
12:
11:
5:
4076:
4066:
4065:
4063:Measure theory
4060:
4043:
4042:
4040:
4039:
4034:
4029:
4024:
4019:
4013:
4011:
4007:
4006:
4003:
4002:
4000:
3999:
3994:
3989:
3984:
3983:
3982:
3972:
3966:
3964:
3955:
3954:
3952:
3951:
3946:
3944:Sard's theorem
3941:
3936:
3931:
3930:
3929:
3927:Lifting theory
3918:
3912:
3908:
3907:
3905:
3904:
3899:
3894:
3889:
3884:
3883:
3882:
3880:FubiniâTonelli
3872:
3867:
3862:
3861:
3860:
3855:
3850:
3842:
3841:
3840:
3835:
3830:
3822:
3816:
3814:
3808:
3807:
3805:
3804:
3799:
3794:
3789:
3784:
3779:
3774:
3768:
3763:
3762:
3761:
3759:in probability
3756:
3746:
3741:
3736:
3730:
3729:
3728:
3723:
3718:
3707:
3705:
3701:
3700:
3698:
3697:
3692:
3687:
3682:
3677:
3672:
3671:
3670:
3660:
3655:
3654:
3653:
3643:
3638:
3633:
3628:
3623:
3618:
3613:
3608:
3602:
3600:
3594:
3593:
3591:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3539:
3538:
3533:
3528:
3518:
3513:
3508:
3503:
3493:
3488:
3483:
3478:
3473:
3468:
3466:Locally finite
3463:
3453:
3448:
3443:
3438:
3433:
3428:
3418:
3413:
3408:
3403:
3398:
3393:
3388:
3383:
3378:
3372:
3370:
3363:
3362:
3360:
3359:
3354:
3349:
3344:
3339:
3338:
3337:
3327:
3322:
3314:
3309:
3308:
3307:
3297:
3292:
3291:
3290:
3280:
3275:
3270:
3269:
3268:
3258:
3253:
3248:
3242:
3240:
3236:
3235:
3233:
3232:
3223:
3222:
3221:
3211:
3206:
3198:
3193:
3183:
3181:
3180:Basic concepts
3177:
3176:
3173:Measure theory
3169:
3168:
3161:
3154:
3146:
3140:
3139:
3129:(3): 214â218.
3114:
3111:
3108:
3107:
3088:
3063:
3062:
3060:
3057:
3056:
3055:
3049:
3041:
3038:
3017:
3012:
3009:
2997:geometric mean
2969:
2966:
2963:
2927:
2901:
2898:
2893:
2869:
2866:
2861: on
2858:
2855:
2852:
2848:
2844:
2841:
2838:
2835:
2831:
2825:
2822:
2818:
2811:
2806:
2802:
2797:
2792:
2788:
2778:
2773: on
2770:
2767:
2764:
2760:
2756:
2753:
2750:
2747:
2743:
2737:
2733:
2726:
2721:
2717:
2712:
2707:
2703:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2635:
2632:
2629:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2489:
2484:
2481:
2476:
2471:
2466:
2463:
2458:
2453:
2450:
2438:
2435:
2422:
2419:
2399:
2396:
2350:
2347:
2342:
2309:
2306:
2266:
2261:
2258:
2253:
2248:
2245:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2153:
2148:
2145:
2140:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2083:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2017:
2014:
2009:
2004:
2001:
1998:
1995:
1992:
1989:
1977:
1974:
1930:
1927:
1906:
1903:
1899:
1895:
1892:
1889:
1886:
1881:
1877:
1873:
1870:
1866:
1862:
1859:
1856:
1853:
1850:
1836:quotient group
1821:
1818:
1813:
1808:
1805:
1801:
1797:
1794:
1791:
1788:
1783:
1779:
1758:
1753:
1748:
1743:
1740:
1736:
1732:
1729:
1726:
1722:
1719:
1694:
1690:
1686:
1683:
1679:
1675:
1672:
1669:
1665:
1644:
1641:
1638:
1614:
1610:
1606:
1603:
1572:
1552:
1549:
1546:
1543:
1540:
1520:
1515:
1511:
1507:
1504:
1487:, the highest
1470:
1467:
1462:
1439:
1435:
1432:
1427:
1424:
1419:
1413:
1388:
1368:
1363:
1359:
1338:
1319:
1315:
1312:
1307:
1304:
1299:
1293:
1265:
1262:
1239:
1236:
1216:
1213:
1193:
1189:
1185:
1182:
1177:
1173:
1169:
1164:
1160:
1156:
1151:
1147:
1142:
1116:
1113:
1108:
1103:
1100:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
992:
989:
986:
983:
980:
977:
973:
969:
965:
962:
959:
956:
953:
950:
947:
944:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
866:
863:
858:
830:
827:
824:
821:
801:
798:
795:
792:
783:is zero; when
768:
765:
762:
759:
756:
753:
750:
747:
726:
721:
717:
713:
689:
686:
641:
638:
633:
619:
616:
604:
601:
598:
575:
570:
566:
560:
554:
550:
546:
542:
539:
502:
499:
494:
464:
460:
457:
454:
451:
447:
443:
440:
436:
413:
408:
403:
379:
374:
369:
346:
342:
339:
336:
333:
329:
325:
322:
318:
295:
290:
266:
261:
237:
232:
208:
203:
181:
177:
173:
170:
167:
164:
160:
156:
153:
149:
145:
140:
137:
132:
102:
98:
94:
91:
88:
85:
81:
77:
74:
70:
66:
61:
58:
53:
26:
9:
6:
4:
3:
2:
4075:
4064:
4061:
4059:
4056:
4055:
4053:
4038:
4035:
4033:
4032:Real analysis
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4014:
4012:
4008:
3998:
3995:
3993:
3990:
3988:
3985:
3981:
3978:
3977:
3976:
3973:
3971:
3968:
3967:
3965:
3962:
3956:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3928:
3925:
3924:
3923:
3920:
3919:
3916:
3913:
3911:Other results
3909:
3903:
3900:
3898:
3897:RadonâNikodym
3895:
3893:
3890:
3888:
3885:
3881:
3878:
3877:
3876:
3873:
3871:
3870:Fatou's lemma
3868:
3866:
3863:
3859:
3856:
3854:
3851:
3849:
3846:
3845:
3843:
3839:
3836:
3834:
3831:
3829:
3826:
3825:
3823:
3821:
3818:
3817:
3815:
3813:
3809:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3769:
3767:
3764:
3760:
3757:
3755:
3752:
3751:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:Convergence:
3731:
3727:
3724:
3722:
3719:
3717:
3714:
3713:
3712:
3709:
3708:
3706:
3702:
3696:
3693:
3691:
3688:
3686:
3683:
3681:
3678:
3676:
3673:
3669:
3666:
3665:
3664:
3661:
3659:
3656:
3652:
3649:
3648:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3603:
3601:
3599:
3595:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3537:
3536:Outer regular
3534:
3532:
3531:Inner regular
3529:
3527:
3526:Borel regular
3524:
3523:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3498:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3458:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3423:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3373:
3371:
3369:
3364:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3336:
3333:
3332:
3331:
3328:
3326:
3323:
3321:
3315:
3313:
3310:
3306:
3303:
3302:
3301:
3298:
3296:
3293:
3289:
3286:
3285:
3284:
3281:
3279:
3276:
3274:
3271:
3267:
3264:
3263:
3262:
3259:
3257:
3254:
3252:
3249:
3247:
3244:
3243:
3241:
3237:
3231:
3227:
3224:
3220:
3217:
3216:
3215:
3214:Measure space
3212:
3210:
3207:
3205:
3203:
3199:
3197:
3194:
3192:
3188:
3185:
3184:
3182:
3178:
3174:
3167:
3162:
3160:
3155:
3153:
3148:
3147:
3144:
3136:
3132:
3128:
3124:
3123:
3117:
3116:
3105:
3101:
3097:
3092:
3078:
3074:
3068:
3064:
3053:
3050:
3047:
3044:
3043:
3037:
3035:
3031:
3015:
3010:
3007:
2998:
2994:
2990:
2986:
2981:
2967:
2964:
2961:
2953:
2949:
2945:
2941:
2925:
2917:
2899:
2896:
2880:
2867:
2864:
2853:
2850:
2846:
2842:
2839:
2836:
2833:
2829:
2823:
2820:
2816:
2809:
2804:
2800:
2795:
2790:
2776:
2765:
2762:
2758:
2754:
2751:
2748:
2745:
2741:
2735:
2731:
2724:
2719:
2715:
2710:
2705:
2693:
2674:
2671:
2668:
2662:
2659:
2656:
2653:
2633:
2630:
2627:
2618:
2605:
2599:
2596:
2593:
2590:
2587:
2581:
2578:
2575:
2566:
2563:
2540:
2537:
2534:
2531:
2525:
2522:
2519:
2510:
2507:
2487:
2482:
2479:
2469:
2464:
2461:
2451:
2448:
2434:
2417:
2394:
2386:
2382:
2378:
2374:
2370:
2366:
2348:
2345:
2329:
2327:
2323:
2319:
2315:
2305:
2303:
2299:
2295:
2291:
2287:
2282:
2280:
2264:
2259:
2256:
2246:
2243:
2220:
2217:
2214:
2211:
2208:
2196:
2193:
2190:
2179:
2175:
2171:
2168:of the usual
2167:
2151:
2146:
2143:
2128:
2125:determines a
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2085:
2081:
2077:
2071:
2068:
2065:
2056:
2053:
2050:
2041:
2033:
2015:
2012:
2002:
1996:
1993:
1990:
1973:
1971:
1967:
1962:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1926:
1924:
1920:
1904:
1893:
1890:
1884:
1879:
1875:
1871:
1860:
1857:
1851:
1848:
1841:
1837:
1819:
1816:
1795:
1792:
1786:
1781:
1777:
1756:
1751:
1730:
1727:
1708:
1692:
1673:
1670:
1642:
1639:
1636:
1628:
1612:
1604:
1601:
1593:
1588:
1586:
1570:
1550:
1547:
1544:
1541:
1538:
1518:
1513:
1509:
1505:
1502:
1494:
1490:
1486:
1468:
1465:
1437:
1433:
1430:
1425:
1422:
1411:
1402:
1386:
1366:
1361:
1357:
1349:
1346:
1342:
1336:
1334:
1317:
1313:
1310:
1305:
1302:
1291:
1281:
1279:
1263:
1260:
1253:
1237:
1234:
1214:
1211:
1191:
1187:
1183:
1180:
1175:
1171:
1167:
1162:
1158:
1154:
1149:
1145:
1140:
1132:
1114:
1111:
1101:
1098:
1078:
1072:
1066:
1063:
1057:
1053:
1047:
1044:
1041:
1026:
1023:
1017:
1014:
1006:
990:
984:
975:
971:
954:
951:
945:
942:
934:
930:
908:
905:
899:
893:
887:
881:
878:
875:
869:
864:
861:
846:
844:
828:
825:
822:
819:
799:
796:
793:
790:
782:
766:
760:
757:
754:
748:
745:
724:
719:
715:
711:
703:
687:
684:
675:
673:
669:
665:
661:
657:
639:
636:
615:
602:
599:
596:
589:
573:
568:
548:
540:
537:
530:
526:
522:
518:
500:
497:
482:
481:complex plane
477:
462:
458:
455:
452:
449:
441:
438:
434:
411:
406:
377:
372:
344:
340:
337:
334:
331:
323:
320:
316:
293:
264:
235:
206:
179:
175:
171:
168:
165:
162:
154:
151:
147:
143:
138:
135:
120:
116:
100:
96:
92:
89:
86:
83:
75:
72:
68:
64:
59:
56:
41:
37:
33:
19:
3812:Main results
3548:Set function
3476:Metric outer
3431:Decomposable
3288:Cylinder set
3201:
3126:
3120:
3113:Bibliography
3091:
3080:. Retrieved
3076:
3067:
3029:
2992:
2988:
2982:
2951:
2947:
2943:
2881:
2619:
2440:
2385:log semiring
2330:
2311:
2308:Applications
2283:
2279:Haar measure
1979:
1963:
1932:
1594:, for every
1589:
1484:
1282:
847:
676:
621:
516:
478:
118:
115:real numbers
39:
29:
3772:compact set
3739:of measures
3675:Pushforward
3668:Projections
3658:Logarithmic
3501:Probability
3491:Pre-measure
3273:Borel space
3191:of measures
3077:ncatlab.org
2381:isomorphism
2379:) gives an
2298:ISO 80000-3
1947:denominator
1929:Ratio scale
1917:called the
1627:determinant
1485:ratio scale
1333:total order
32:mathematics
18:Ratio scale
4052:Categories
3744:in measure
3471:Maximising
3441:Equivalent
3335:Vitali set
3082:2020-08-11
3059:References
1970:proportion
1933:Among the
618:Properties
3858:Maharam's
3828:Dominated
3641:Intensity
3636:Hausdorff
3543:Saturated
3461:Invariant
3366:Types of
3325:Ï-algebra
3295:đ-system
3261:Borel set
3256:Baire set
3046:Semifield
2965:×
2854:×
2840:∈
2821:−
2766:×
2752:∈
2657:∩
2631:∪
2470:×
2421:∞
2418:−
2398:∞
2395:−
2383:with the
2346:≥
2247:∈
2203:→
2110:
2104:−
2098:
2072:
2042:μ
2003:⊆
1943:numerator
1923:Lie group
1885:
1872:◃
1852:
1807:→
1787:
1752:×
1742:→
1685:→
1640:×
1605:∈
1542:≤
1506:×
1184:…
1102:∈
1076:⌋
1070:⌊
1067:−
1061:↦
1036:→
1030:∞
988:⌋
982:⌊
979:↦
964:→
958:∞
912:∞
900:∪
888:∪
843:unbounded
749:∈
664:real line
662:from the
597:φ
569:φ
450:∣
442:∈
407:∗
378:∗
338:≥
332:∣
324:∈
169:≥
163:∣
155:∈
136:≥
84:∣
76:∈
3875:Fubini's
3865:Egorov's
3833:Monotone
3792:variable
3770:Random:
3721:Strongly
3646:Lebesgue
3631:Harmonic
3621:Gaussian
3606:Counting
3573:Spectral
3568:Singular
3558:s-finite
3553:Ï-finite
3436:Discrete
3411:Complete
3368:Measures
3342:Null set
3230:function
3040:See also
2365:semiring
2326:measures
2290:decibels
2166:pullback
2032:interval
1939:division
1483:forms a
1331:forms a
1250:and for
927:and the
702:sequence
660:topology
622:The set
588:argument
3787:process
3782:measure
3777:element
3716:Bochner
3690:Trivial
3685:Tangent
3663:Product
3521:Regular
3499:)
3486:Perfect
3459:)
3424:)
3416:Content
3406:Complex
3347:Support
3320:-system
3209:Measure
3098:(1961)
2318:metrics
2127:measure
2034:, then
1966:Eudoxus
1838:by the
1401:integer
1335:but is
3853:Jordan
3838:Vitali
3797:vector
3726:Weakly
3588:Vector
3563:Signed
3516:Random
3457:Quasi-
3446:Finite
3426:Convex
3386:Banach
3376:Atomic
3204:spaces
3189:
2882:Since
2813:
2728:
2437:Square
2363:has a
2324:, and
2302:levels
2030:is an
1585:decade
1531:where
1399:is an
1379:where
1343:. The
1015:excess
1005:excess
656:closed
34:, the
3695:Young
3616:Euler
3611:Dirac
3583:Tight
3511:Radon
3481:Outer
3451:Inner
3401:Brown
3396:Borel
3391:Besov
3381:Baire
2938:is a
2916:group
2914:is a
2322:norms
2314:image
2236:by a
1129:as a
943:floor
933:floor
781:limit
586:with
479:In a
3959:For
3848:Hahn
3704:Maps
3626:Haar
3497:Sub-
3251:Atom
3239:Sets
2946:and
2897:>
2620:The
2480:>
2462:>
2441:Let
2316:for
2257:>
2144:>
2013:>
1945:and
1817:>
1625:the
1563:and
1548:<
1466:>
1434:>
1423:>
1314:>
1303:>
1112:>
1003:and
862:>
823:>
779:the
700:the
637:>
498:>
456:>
426:for
358:and
308:for
221:and
90:>
57:>
3131:doi
3127:188
2987:in
2950:in
2107:log
2095:log
2069:log
1980:If
1495:as
1337:not
654:is
521:ray
392:or
279:or
38:of
36:set
30:In
4054::
3125:.
3075:.
3036:.
2918:,
2320:,
2304:.
2281:.
1961:.
1925:.
1876:GL
1849:SL
1778:GL
1551:10
1510:10
1358:10
1339:a
1280:.
1007:,
935:,
845:.
674:.
603:0.
483:,
121:,
42:,
3495:(
3455:(
3420:(
3318:Ï
3228:/
3202:L
3165:e
3158:t
3151:v
3137:.
3133::
3085:.
3030:H
3016:,
3011:y
3008:x
2993:L
2989:Q
2968:H
2962:L
2952:Q
2948:H
2944:L
2926:Q
2900:0
2892:R
2868:.
2865:H
2857:}
2851:,
2847:}
2843:R
2837:a
2834::
2830:)
2824:a
2817:e
2810:,
2805:a
2801:e
2796:(
2791:{
2787:{
2777:L
2769:}
2763:,
2759:}
2755:R
2749:a
2746::
2742:)
2736:a
2732:e
2725:,
2720:a
2716:e
2711:(
2706:{
2702:{
2678:)
2675:1
2672:,
2669:1
2666:(
2663:=
2660:H
2654:L
2634:H
2628:L
2606:.
2603:}
2600:1
2597:=
2594:y
2591:x
2588::
2585:)
2582:y
2579:,
2576:x
2573:(
2570:{
2567:=
2564:H
2544:}
2541:y
2538:=
2535:x
2532::
2529:)
2526:y
2523:,
2520:x
2517:(
2514:{
2511:=
2508:L
2488:,
2483:0
2475:R
2465:0
2457:R
2452:=
2449:Q
2349:0
2341:R
2265:,
2260:0
2252:R
2244:z
2224:]
2221:z
2218:b
2215:,
2212:z
2209:a
2206:[
2200:]
2197:b
2194:,
2191:a
2188:[
2152:,
2147:0
2139:R
2113:a
2101:b
2092:=
2089:)
2086:a
2082:/
2078:b
2075:(
2066:=
2063:)
2060:]
2057:b
2054:,
2051:a
2048:[
2045:(
2016:0
2008:R
2000:]
1997:b
1994:,
1991:a
1988:[
1905:,
1902:)
1898:R
1894:,
1891:n
1888:(
1880:+
1869:)
1865:R
1861:,
1858:n
1855:(
1820:0
1812:R
1804:)
1800:R
1796:,
1793:n
1790:(
1782:+
1757:.
1747:R
1739:)
1735:R
1731:,
1728:n
1725:(
1721:L
1718:G
1693:.
1689:R
1682:)
1678:R
1674:,
1671:n
1668:(
1664:M
1643:n
1637:n
1613:,
1609:N
1602:n
1571:b
1545:a
1539:1
1519:,
1514:b
1503:a
1469:0
1461:R
1438:)
1431:,
1426:0
1418:R
1412:(
1387:n
1367:,
1362:n
1318:)
1311:,
1306:0
1298:R
1292:(
1264:,
1261:x
1238:,
1235:x
1215:,
1212:x
1192:,
1188:]
1181:,
1176:2
1172:n
1168:,
1163:1
1159:n
1155:;
1150:0
1146:n
1141:[
1115:0
1107:R
1099:x
1079:,
1073:x
1064:x
1058:x
1054:,
1051:)
1048:1
1045:,
1042:0
1039:(
1033:)
1027:,
1024:1
1021:[
1018::
991:,
985:x
976:x
972:,
968:N
961:)
955:,
952:1
949:[
946::
915:)
909:,
906:1
903:(
897:}
894:1
891:{
885:)
882:1
879:,
876:0
873:(
870:=
865:0
857:R
829:,
826:1
820:x
800:,
797:1
794:=
791:x
767:,
764:)
761:1
758:,
755:0
752:(
746:x
725:}
720:n
716:x
712:{
688:,
685:x
640:0
632:R
600:=
574:,
565:i
559:e
553:|
549:z
545:|
541:=
538:z
501:0
493:R
463:}
459:0
453:x
446:R
439:x
435:{
412:+
402:R
373:+
368:R
345:}
341:0
335:x
328:R
321:x
317:{
294:+
289:R
265:+
260:R
236:+
231:R
207:+
202:R
180:,
176:}
172:0
166:x
159:R
152:x
148:{
144:=
139:0
131:R
101:,
97:}
93:0
87:x
80:R
73:x
69:{
65:=
60:0
52:R
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.