36:
3017:
1191:
3188:
865:
372:
1407:
3409:
1737:
2636:
2719:
2296:
1330:
2438:
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3084:
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1056:
2553:
2497:
680:
1085:
612:
984:
464:
237:
1775:
924:
1089:
3089:
1545:
764:
1997:
1876:
3498:
3566:
One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets
1334:
3527:
3659:
2818:
2210:
752:
2066:
3615:
3320:
3315:
3226:
2774:
1950:
1829:
1254:
951:
3739:
3562:
3429:
2157:
1628:
726:
703:
2183:
1585:
1280:
3453:
1447:
3703:
3683:
3290:
2913:
2039:
1924:
1896:
1803:
1605:
1565:
1499:
1467:
1427:
1211:
523:
431:
270:
3246:
2350:
2129:
2107:
1655:
2327:
579:
559:
161:
499:
392:) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity.
3086:
is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality
2215:
1659:
239:
If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of
2558:
2641:
53:
100:
72:
617:
1292:
79:
2354:
255:(the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an
86:
17:
2823:
68:
3024:
996:
3893:
119:
2502:
2446:
3774:
3569:
1061:
588:
963:
443:
174:
1742:
870:
57:
1512:
3012:{\displaystyle \mu (A)={\begin{cases}1&{\mbox{ if }}0\in A\\0&{\mbox{ if }}0\notin A.\end{cases}}}
1955:
1834:
3928:
3792:
3462:
93:
3503:
2013:
3620:
2779:
2188:
731:
502:
2950:
2738:
Note that modularity has a different and unrelated meaning in the context of complex functions; see
2047:
3825:
2730:
2638:
where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
3298:
3211:
2753:
1929:
1808:
3923:
3819:
3708:
2724:
1232:
929:
46:
3532:
1186:{\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}),}
3183:{\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})}
758:
141:
3414:
2133:
1610:
860:{\displaystyle \mu \left(\bigcup _{n=1}^{N}A_{n}\right)=\sum _{n=1}^{N}\mu \left(A_{n}\right)}
708:
685:
3801:
3194:
2162:
1570:
1470:
377:
367:{\textstyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}).}
1262:
3438:
3272:
An example of an additive function which is not σ-additive is obtained by considering
2078:
2007:
1432:
1402:{\displaystyle \mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G),}
3688:
3668:
3275:
2898:
2024:
1909:
1881:
1788:
1590:
1550:
1484:
1452:
1412:
1196:
508:
416:
8:
3762:
3754:
3807:
3231:
2332:
2111:
2089:
1637:
3848:
2312:
564:
544:
264:
164:
3404:{\displaystyle \mu (A)=\lim _{k\to \infty }{\frac {1}{k}}\cdot \lambda (A\cap (0,k)),}
469:
146:
3899:
3889:
1257:
247:
is a finite number) equals the sum of its values on the sets. Therefore, an additive
3766:
3432:
3830:
3813:
3770:
3662:
3198:
2042:
1221:. Every 𝜎-additive function is additive but not vice versa, as shown below.
3917:
987:
438:
3903:
1732:{\displaystyle \mu (A)=\mu (A\cup \varnothing )=\mu (A)+\mu (\varnothing ).}
3786:
3758:
3456:
2739:
2019:
1283:
582:
434:
248:
168:
3883:
3293:
2920:
376:
Additivity and sigma-additivity are particularly important properties of
133:
2631:{\displaystyle A\cup B=(A\cap B)\cup (A\setminus B)\cup (B\setminus A),}
3844:
2916:
2714:{\displaystyle \mu (A\setminus B)+\mu (B\setminus A)+2\mu (A\cap B).}
1777:
then this equality can be satisfied only by plus or minus infinity.
682:
A consequence of this is that an additive function cannot take both
35:
3835:
3761:). For sigma-additivity, one needs in addition that the concept of
3249:
1287:
991:
2306:
and the argument below proves that additivity implies modularity.
408:
3750:
3267:
2291:{\displaystyle \phi (A\cup B)+\phi (A\cap B)=\phi (A)+\phi (B)}
389:
381:
163:
mapping sets to numbers, with the property that its value on a
3749:
One may define additive functions with values in any additive
1325:{\textstyle {\mathcal {G}}\subseteq {\mathcal {A}}\cap \tau ,}
3705:
applied to any of the individual sets is zero, so the sum of
380:. They are abstractions of how intuitive properties of size (
3208:
is defined to be a finitely additive set function that maps
2433:{\displaystyle \mu (A\cup B)+\mu (A\cap B)=\mu (A)+\mu (B).}
3005:
1473:(with respect to compact sets) then it is τ-additive.
385:
171:
sets equals the sum of its values on these sets, namely,
3885:
Theory of charges: a study of finitely additive measures
3881:
263:
is a function that has the additivity property even for
3804: – Generalization of mass, length, area and volume
3797:
Pages displaying short descriptions of redirect targets
3773:. Another example, also from quantum mechanics, is the
2984:
2959:
1295:
1235:
1065:
967:
932:
592:
447:
273:
177:
149:
3838:– The set of bounded charges on a given sigma-algebra
3711:
3691:
3671:
3623:
3572:
3535:
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3465:
3441:
3417:
3323:
3301:
3278:
3234:
3214:
3092:
3027:
2929:
2901:
2895:
An example of a 𝜎-additive function is the function
2826:
2782:
2756:
2644:
2561:
2505:
2449:
2357:
2335:
2315:
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2191:
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1932:
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1199:
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767:
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711:
688:
620:
591:
567:
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419:
3843:
This article incorporates material from additive on
3816: – Generalized notion of measure in mathematics
2881:{\displaystyle \mu (B\setminus A)=\mu (B)-\mu (A).}
60:. Unsourced material may be challenged and removed.
3795: – Theorem extending pre-measures to measures
3733:
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366:
231:
155:
3882:Bhaskara Rao, K. P. S.; Bhaskara Rao, M. (1983).
3268:An additive function which is not σ-additive
3079:{\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots }
1051:{\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots }
3915:
3849:Creative Commons Attribution/Share-Alike License
3822: – Set-to-real map with diminishing returns
3507:
3442:
3340:
1365:
3741:is also zero, which proves the counterexample.
3769:are sigma-additive functions with values in a
2548:{\displaystyle B=(A\cap B)\cup (B\setminus A)}
2492:{\displaystyle A=(A\cap B)\cup (A\setminus B)}
1481:Useful properties of an additive set function
675:{\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}
409:Additive (or finitely additive) set functions
3201:for more examples of 𝜎-additive functions.
1229:Suppose that in addition to a sigma algebra
1080:{\displaystyle \scriptstyle {\mathcal {A}},}
607:{\displaystyle \scriptstyle {\mathcal {A}},}
401:is equivalent to additive set function; see
1224:
979:{\displaystyle \scriptstyle {\mathcal {A}}}
955:
459:{\displaystyle \scriptstyle {\mathcal {A}}}
232:{\textstyle \mu (A\cup B)=\mu (A)+\mu (B).}
3303:
3260:to mean its range is a bounded subset of
1770:{\displaystyle \mu (\varnothing )\neq 0,}
919:{\displaystyle A_{1},A_{2},\ldots ,A_{N}}
397:
120:Learn how and when to remove this message
3685:applied to the union is then one, while
3292:, defined over the Lebesgue sets of the
1634:: additivity implies that for every set
14:
3916:
3888:. London: Academic Press. p. 35.
3862:
3765:be defined on that set. For example,
1540:{\displaystyle \mu (\varnothing )=0,}
1504:
1992:{\displaystyle \mu (A)\geq \mu (B).}
1871:{\displaystyle \mu (A)\leq \mu (B).}
761:that an additive function satisfies
58:adding citations to reliable sources
29:
3493:{\displaystyle 0\leq \mu (A)\leq 1}
2723:However, the related properties of
24:
3744:
3516:
3350:
3256:charges, where we say a charge is
3156:
3117:
2735:are not equivalent to each other.
2194:
2053:
1617:
1574:
1377:
1351:
1308:
1298:
1238:
1156:
1117:
1068:
970:
935:
741:
735:
715:
692:
595:
485:
479:
450:
337:
298:
25:
3940:
3810: – Concept in measure theory
3522:{\displaystyle \sup A<\infty }
2836:
2745:
2675:
2654:
2616:
2598:
2536:
2480:
1752:
1720:
1690:
1522:
3775:positive operator-valued measure
34:
3661:The union of these sets is the
3654:{\displaystyle n=0,1,2,\ldots }
2813:{\displaystyle \mu (B)-\mu (A)}
2205:{\displaystyle {\mathcal {S}},}
1780:
747:{\displaystyle \infty -\infty }
402:
45:needs additional citations for
3875:
3847:, which is licensed under the
3728:
3715:
3604:
3586:
3545:
3539:
3481:
3475:
3395:
3392:
3380:
3371:
3347:
3333:
3327:
3177:
3164:
2939:
2933:
2872:
2866:
2857:
2851:
2842:
2830:
2807:
2801:
2792:
2786:
2705:
2693:
2681:
2669:
2660:
2648:
2622:
2610:
2604:
2592:
2586:
2574:
2542:
2530:
2524:
2512:
2486:
2474:
2468:
2456:
2424:
2418:
2409:
2403:
2394:
2382:
2373:
2361:
2285:
2279:
2270:
2264:
2255:
2243:
2234:
2222:
2061:{\displaystyle {\mathcal {S}}}
1983:
1977:
1968:
1962:
1862:
1856:
1847:
1841:
1755:
1749:
1723:
1717:
1708:
1702:
1693:
1681:
1672:
1666:
1587:to all sets in its domain, or
1525:
1519:
1393:
1387:
1177:
1164:
728:as values, for the expression
666:
660:
651:
645:
636:
624:
488:
473:
358:
345:
253:finitely additive set function
223:
217:
208:
202:
193:
181:
13:
1:
3855:
3789: – Z-module homomorphism
3610:{\displaystyle A_{n}=[n,n+1)}
2298:The above property is called
2001:
1476:
1449:-additive. In particular, if
1225:τ-additive set functions
1058:of pairwise disjoint sets in
956:σ-additive set functions
69:"Sigma-additive set function"
3310:{\displaystyle \mathbb {R} }
3221:{\displaystyle \varnothing }
2769:{\displaystyle A\subseteq B}
1945:{\displaystyle A\subseteq B}
1824:{\displaystyle A\subseteq B}
1249:{\textstyle {\mathcal {A}},}
946:{\textstyle {\mathcal {A}}.}
261:σ-additive set function
7:
3780:
3734:{\displaystyle \mu (A_{n})}
2890:
1630:to all sets in its domain.
10:
3945:
3557:{\displaystyle \mu (A)=0.}
2014:Valuation (measure theory)
2011:
2005:
503:extended real number line
3870:Measure Theory, Volume 4
3826:Subadditive set function
3424:{\displaystyle \lambda }
2152:{\displaystyle A\cup B,}
1623:{\displaystyle -\infty }
721:{\displaystyle +\infty }
698:{\displaystyle -\infty }
3872:, Torres Fremlin, 2003.
3820:Submodular set function
3793:Hahn–Kolmogorov theorem
2178:{\displaystyle A\cap B}
1580:{\displaystyle \infty }
1501:include the following.
3735:
3699:
3679:
3655:
3611:
3558:
3523:
3494:
3449:
3425:
3405:
3311:
3286:
3252:for information about
3242:
3222:
3184:
3160:
3121:
3080:
3013:
2909:
2882:
2814:
2770:
2715:
2632:
2549:
2493:
2434:
2346:
2323:
2292:
2206:
2179:
2153:
2125:
2103:
2062:
2035:
1993:
1946:
1920:
1892:
1872:
1825:
1799:
1771:
1733:
1651:
1624:
1601:
1581:
1561:
1541:
1495:
1463:
1443:
1423:
1403:
1326:
1276:
1275:{\displaystyle \tau .}
1250:
1207:
1187:
1160:
1121:
1081:
1052:
980:
947:
920:
861:
835:
796:
759:mathematical induction
748:
722:
699:
676:
608:
575:
555:
519:
495:
460:
427:
368:
341:
302:
233:
157:
3802:Measure (mathematics)
3736:
3700:
3680:
3656:
3612:
3559:
3524:
3495:
3450:
3448:{\displaystyle \lim }
3426:
3406:
3312:
3287:
3243:
3223:
3185:
3140:
3101:
3081:
3014:
2910:
2883:
2815:
2771:
2716:
2633:
2550:
2494:
2435:
2347:
2324:
2293:
2207:
2180:
2154:
2126:
2104:
2063:
2036:
1994:
1947:
1921:
1902:monotone set function
1893:
1873:
1826:
1800:
1772:
1734:
1652:
1625:
1602:
1582:
1562:
1542:
1496:
1464:
1444:
1442:{\displaystyle \tau }
1424:
1404:
1327:
1286:family of measurable
1277:
1251:
1208:
1188:
1140:
1101:
1082:
1053:
981:
948:
921:
862:
815:
776:
749:
723:
700:
677:
609:
576:
556:
520:
496:
461:
428:
369:
321:
282:
243:disjoint sets (where
234:
158:
138:additive set function
18:Additive set function
3709:
3698:{\displaystyle \mu }
3689:
3678:{\displaystyle \mu }
3669:
3621:
3570:
3533:
3504:
3463:
3439:
3415:
3321:
3299:
3285:{\displaystyle \mu }
3276:
3232:
3212:
3090:
3025:
2927:
2908:{\displaystyle \mu }
2899:
2824:
2780:
2754:
2642:
2559:
2503:
2447:
2355:
2333:
2313:
2216:
2189:
2163:
2134:
2112:
2090:
2072:modular set function
2048:
2034:{\displaystyle \mu }
2025:
2008:Valuation (geometry)
1956:
1930:
1926:is non-positive and
1919:{\displaystyle \mu }
1910:
1891:{\displaystyle \mu }
1882:
1835:
1809:
1805:is non-negative and
1798:{\displaystyle \mu }
1789:
1743:
1660:
1638:
1611:
1600:{\displaystyle \mu }
1591:
1571:
1560:{\displaystyle \mu }
1551:
1513:
1494:{\displaystyle \mu }
1485:
1462:{\displaystyle \mu }
1453:
1433:
1422:{\displaystyle \mu }
1413:
1335:
1293:
1263:
1233:
1206:{\displaystyle \mu }
1197:
1090:
1062:
997:
964:
930:
871:
765:
732:
709:
686:
618:
589:
565:
545:
518:{\displaystyle \mu }
509:
470:
444:
426:{\displaystyle \mu }
417:
398:modular set function
271:
267:many sets, that is,
175:
147:
54:improve this article
3763:limit of a sequence
3757:or more commonly a
3929:Additive functions
3731:
3695:
3675:
3651:
3607:
3554:
3519:
3490:
3445:
3421:
3401:
3354:
3307:
3282:
3241:{\displaystyle 0.}
3238:
3218:
3180:
3076:
3009:
3004:
2988:
2963:
2905:
2878:
2810:
2766:
2711:
2628:
2545:
2489:
2430:
2345:{\displaystyle B,}
2342:
2319:
2288:
2202:
2175:
2149:
2124:{\displaystyle B,}
2121:
2102:{\displaystyle A,}
2099:
2058:
2031:
1989:
1942:
1916:
1888:
1868:
1821:
1795:
1767:
1729:
1650:{\displaystyle A,}
1647:
1620:
1597:
1577:
1557:
1537:
1505:Value of empty set
1491:
1459:
1439:
1419:
1399:
1383:
1322:
1272:
1246:
1215:countably additive
1203:
1183:
1077:
1076:
1048:
976:
975:
943:
916:
857:
744:
718:
695:
672:
604:
603:
571:
551:
515:
491:
456:
455:
423:
364:
265:countably infinite
259:number of sets. A
229:
153:
3767:spectral measures
3753:(for example any
3363:
3339:
2987:
2962:
2915:defined over the
2820:is defined, then
2322:{\displaystyle A}
1364:
926:disjoint sets in
757:One can prove by
574:{\displaystyle B}
554:{\displaystyle A}
537:finitely additive
251:is also called a
156:{\textstyle \mu }
130:
129:
122:
104:
16:(Redirected from
3936:
3908:
3907:
3879:
3873:
3866:
3808:σ-finite measure
3798:
3740:
3738:
3737:
3732:
3727:
3726:
3704:
3702:
3701:
3696:
3684:
3682:
3681:
3676:
3660:
3658:
3657:
3652:
3616:
3614:
3613:
3608:
3582:
3581:
3563:
3561:
3560:
3555:
3528:
3526:
3525:
3520:
3499:
3497:
3496:
3491:
3454:
3452:
3451:
3446:
3433:Lebesgue measure
3430:
3428:
3427:
3422:
3410:
3408:
3407:
3402:
3364:
3356:
3353:
3316:
3314:
3313:
3308:
3306:
3291:
3289:
3288:
3283:
3247:
3245:
3244:
3239:
3227:
3225:
3224:
3219:
3189:
3187:
3186:
3181:
3176:
3175:
3159:
3154:
3136:
3132:
3131:
3130:
3120:
3115:
3085:
3083:
3082:
3077:
3069:
3068:
3050:
3049:
3037:
3036:
3018:
3016:
3015:
3010:
3008:
3007:
2989:
2985:
2964:
2960:
2914:
2912:
2911:
2906:
2887:
2885:
2884:
2879:
2819:
2817:
2816:
2811:
2775:
2773:
2772:
2767:
2720:
2718:
2717:
2712:
2637:
2635:
2634:
2629:
2554:
2552:
2551:
2546:
2498:
2496:
2495:
2490:
2439:
2437:
2436:
2431:
2351:
2349:
2348:
2343:
2328:
2326:
2325:
2320:
2304:
2303:
2297:
2295:
2294:
2289:
2211:
2209:
2208:
2203:
2198:
2197:
2185:are elements of
2184:
2182:
2181:
2176:
2158:
2156:
2155:
2150:
2130:
2128:
2127:
2122:
2108:
2106:
2105:
2100:
2083:
2082:
2074:
2073:
2067:
2065:
2064:
2059:
2057:
2056:
2040:
2038:
2037:
2032:
1998:
1996:
1995:
1990:
1951:
1949:
1948:
1943:
1925:
1923:
1922:
1917:
1906:. Similarly, If
1904:
1903:
1897:
1895:
1894:
1889:
1877:
1875:
1874:
1869:
1830:
1828:
1827:
1822:
1804:
1802:
1801:
1796:
1776:
1774:
1773:
1768:
1738:
1736:
1735:
1730:
1656:
1654:
1653:
1648:
1629:
1627:
1626:
1621:
1606:
1604:
1603:
1598:
1586:
1584:
1583:
1578:
1566:
1564:
1563:
1558:
1546:
1544:
1543:
1538:
1500:
1498:
1497:
1492:
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1382:
1381:
1380:
1360:
1356:
1355:
1354:
1331:
1329:
1328:
1323:
1312:
1311:
1302:
1301:
1281:
1279:
1278:
1273:
1255:
1253:
1252:
1247:
1242:
1241:
1212:
1210:
1209:
1204:
1192:
1190:
1189:
1184:
1176:
1175:
1159:
1154:
1136:
1132:
1131:
1130:
1120:
1115:
1086:
1084:
1083:
1078:
1072:
1071:
1057:
1055:
1054:
1049:
1041:
1040:
1022:
1021:
1009:
1008:
985:
983:
982:
977:
974:
973:
952:
950:
949:
944:
939:
938:
925:
923:
922:
917:
915:
914:
896:
895:
883:
882:
866:
864:
863:
858:
856:
852:
851:
834:
829:
811:
807:
806:
805:
795:
790:
753:
751:
750:
745:
727:
725:
724:
719:
704:
702:
701:
696:
681:
679:
678:
673:
613:
611:
610:
605:
599:
598:
580:
578:
577:
572:
560:
558:
557:
552:
539:
538:
531:
530:
524:
522:
521:
516:
505:). The function
500:
498:
497:
494:{\displaystyle }
492:
465:
463:
462:
457:
454:
453:
432:
430:
429:
424:
373:
371:
370:
365:
357:
356:
340:
335:
317:
313:
312:
311:
301:
296:
238:
236:
235:
230:
162:
160:
159:
154:
125:
118:
114:
111:
105:
103:
62:
38:
30:
27:Mapping function
21:
3944:
3943:
3939:
3938:
3937:
3935:
3934:
3933:
3914:
3913:
3912:
3911:
3896:
3880:
3876:
3867:
3863:
3858:
3796:
3783:
3747:
3745:Generalizations
3722:
3718:
3710:
3707:
3706:
3690:
3687:
3686:
3670:
3667:
3666:
3622:
3619:
3618:
3577:
3573:
3571:
3568:
3567:
3534:
3531:
3530:
3505:
3502:
3501:
3464:
3461:
3460:
3459:. It satisfies
3440:
3437:
3436:
3416:
3413:
3412:
3355:
3343:
3322:
3319:
3318:
3317:by the formula
3302:
3300:
3297:
3296:
3277:
3274:
3273:
3270:
3233:
3230:
3229:
3213:
3210:
3209:
3171:
3167:
3155:
3144:
3126:
3122:
3116:
3105:
3100:
3096:
3091:
3088:
3087:
3064:
3060:
3045:
3041:
3032:
3028:
3026:
3023:
3022:
3003:
3002:
2983:
2981:
2975:
2974:
2958:
2956:
2946:
2945:
2928:
2925:
2924:
2900:
2897:
2896:
2893:
2825:
2822:
2821:
2781:
2778:
2777:
2755:
2752:
2751:
2748:
2643:
2640:
2639:
2560:
2557:
2556:
2504:
2501:
2500:
2448:
2445:
2444:
2356:
2353:
2352:
2334:
2331:
2330:
2314:
2311:
2310:
2301:
2300:
2217:
2214:
2213:
2193:
2192:
2190:
2187:
2186:
2164:
2161:
2160:
2135:
2132:
2131:
2113:
2110:
2109:
2091:
2088:
2087:
2080:
2079:
2071:
2070:
2052:
2051:
2049:
2046:
2045:
2026:
2023:
2022:
2016:
2010:
2004:
1957:
1954:
1953:
1931:
1928:
1927:
1911:
1908:
1907:
1901:
1900:
1883:
1880:
1879:
1836:
1833:
1832:
1810:
1807:
1806:
1790:
1787:
1786:
1783:
1744:
1741:
1740:
1661:
1658:
1657:
1639:
1636:
1635:
1612:
1609:
1608:
1592:
1589:
1588:
1572:
1569:
1568:
1552:
1549:
1548:
1514:
1511:
1510:
1507:
1486:
1483:
1482:
1479:
1454:
1451:
1450:
1434:
1431:
1430:
1414:
1411:
1410:
1376:
1375:
1368:
1350:
1349:
1345:
1341:
1336:
1333:
1332:
1307:
1306:
1297:
1296:
1294:
1291:
1290:
1264:
1261:
1260:
1237:
1236:
1234:
1231:
1230:
1227:
1198:
1195:
1194:
1171:
1167:
1155:
1144:
1126:
1122:
1116:
1105:
1100:
1096:
1091:
1088:
1087:
1067:
1066:
1063:
1060:
1059:
1036:
1032:
1017:
1013:
1004:
1000:
998:
995:
994:
990:. If for every
969:
968:
965:
962:
961:
958:
934:
933:
931:
928:
927:
910:
906:
891:
887:
878:
874:
872:
869:
868:
847:
843:
839:
830:
819:
801:
797:
791:
780:
775:
771:
766:
763:
762:
733:
730:
729:
710:
707:
706:
687:
684:
683:
619:
616:
615:
594:
593:
590:
587:
586:
566:
563:
562:
546:
543:
542:
536:
535:
528:
527:
510:
507:
506:
471:
468:
467:
466:with values in
449:
448:
445:
442:
441:
439:algebra of sets
418:
415:
414:
411:
352:
348:
336:
325:
307:
303:
297:
286:
281:
277:
272:
269:
268:
176:
173:
172:
148:
145:
144:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3942:
3932:
3931:
3926:
3924:Measure theory
3910:
3909:
3894:
3874:
3868:D. H. Fremlin
3860:
3859:
3857:
3854:
3840:
3839:
3833:
3828:
3823:
3817:
3814:Signed measure
3811:
3805:
3799:
3790:
3782:
3779:
3771:Banach algebra
3746:
3743:
3730:
3725:
3721:
3717:
3714:
3694:
3674:
3663:positive reals
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3580:
3576:
3553:
3550:
3547:
3544:
3541:
3538:
3518:
3515:
3512:
3509:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3444:
3420:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3362:
3359:
3352:
3349:
3346:
3342:
3338:
3335:
3332:
3329:
3326:
3305:
3281:
3269:
3266:
3237:
3217:
3199:signed measure
3179:
3174:
3170:
3166:
3163:
3158:
3153:
3150:
3147:
3143:
3139:
3135:
3129:
3125:
3119:
3114:
3111:
3108:
3104:
3099:
3095:
3075:
3072:
3067:
3063:
3059:
3056:
3053:
3048:
3044:
3040:
3035:
3031:
3006:
3001:
2998:
2995:
2992:
2986: if
2982:
2980:
2977:
2976:
2973:
2970:
2967:
2961: if
2957:
2955:
2952:
2951:
2949:
2944:
2941:
2938:
2935:
2932:
2904:
2892:
2889:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2765:
2762:
2759:
2747:
2746:Set difference
2744:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2341:
2338:
2318:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2201:
2196:
2174:
2171:
2168:
2148:
2145:
2142:
2139:
2120:
2117:
2098:
2095:
2055:
2043:family of sets
2030:
2003:
2000:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1941:
1938:
1935:
1915:
1887:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1820:
1817:
1814:
1794:
1782:
1779:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1646:
1643:
1619:
1616:
1596:
1576:
1556:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1506:
1503:
1490:
1478:
1475:
1458:
1438:
1418:
1398:
1395:
1392:
1389:
1386:
1379:
1374:
1371:
1367:
1363:
1359:
1353:
1348:
1344:
1340:
1321:
1318:
1315:
1310:
1305:
1300:
1271:
1268:
1245:
1240:
1226:
1223:
1220:
1216:
1213:is said to be
1202:
1182:
1179:
1174:
1170:
1166:
1163:
1158:
1153:
1150:
1147:
1143:
1139:
1135:
1129:
1125:
1119:
1114:
1111:
1108:
1104:
1099:
1095:
1075:
1070:
1047:
1044:
1039:
1035:
1031:
1028:
1025:
1020:
1016:
1012:
1007:
1003:
988:σ-algebra
972:
957:
954:
942:
937:
913:
909:
905:
902:
899:
894:
890:
886:
881:
877:
855:
850:
846:
842:
838:
833:
828:
825:
822:
818:
814:
810:
804:
800:
794:
789:
786:
783:
779:
774:
770:
754:is undefined.
743:
740:
737:
717:
714:
694:
691:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
602:
597:
570:
550:
541:, if whenever
514:
490:
487:
484:
481:
478:
475:
452:
437:defined on an
422:
410:
407:
363:
360:
355:
351:
347:
344:
339:
334:
331:
328:
324:
320:
316:
310:
306:
300:
295:
292:
289:
285:
280:
276:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
152:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
3941:
3930:
3927:
3925:
3922:
3921:
3919:
3905:
3901:
3897:
3895:0-12-095780-9
3891:
3887:
3886:
3878:
3871:
3865:
3861:
3853:
3852:
3850:
3846:
3837:
3834:
3832:
3829:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3794:
3791:
3788:
3785:
3784:
3778:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3742:
3723:
3719:
3712:
3692:
3672:
3664:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3601:
3598:
3595:
3592:
3589:
3583:
3578:
3574:
3564:
3551:
3548:
3542:
3536:
3513:
3510:
3487:
3484:
3478:
3472:
3469:
3466:
3458:
3434:
3418:
3398:
3389:
3386:
3383:
3377:
3374:
3368:
3365:
3360:
3357:
3344:
3336:
3330:
3324:
3295:
3279:
3265:
3263:
3259:
3255:
3251:
3235:
3215:
3207:
3202:
3200:
3196:
3191:
3172:
3168:
3161:
3151:
3148:
3145:
3141:
3137:
3133:
3127:
3123:
3112:
3109:
3106:
3102:
3097:
3093:
3073:
3070:
3065:
3061:
3057:
3054:
3051:
3046:
3042:
3038:
3033:
3029:
3019:
2999:
2996:
2993:
2990:
2978:
2971:
2968:
2965:
2953:
2947:
2942:
2936:
2930:
2923:, such that
2922:
2918:
2902:
2888:
2875:
2869:
2863:
2860:
2854:
2848:
2845:
2839:
2833:
2827:
2804:
2798:
2795:
2789:
2783:
2763:
2760:
2757:
2743:
2741:
2736:
2734:
2733:
2732:subadditivity
2728:
2727:
2726:submodularity
2721:
2708:
2702:
2699:
2696:
2690:
2687:
2684:
2678:
2672:
2666:
2663:
2657:
2651:
2645:
2625:
2619:
2613:
2607:
2601:
2595:
2589:
2583:
2580:
2577:
2571:
2568:
2565:
2562:
2539:
2533:
2527:
2521:
2518:
2515:
2509:
2506:
2483:
2477:
2471:
2465:
2462:
2459:
2453:
2450:
2442:
2427:
2421:
2415:
2412:
2406:
2400:
2397:
2391:
2388:
2385:
2379:
2376:
2370:
2367:
2364:
2358:
2339:
2336:
2316:
2307:
2305:
2282:
2276:
2273:
2267:
2261:
2258:
2252:
2249:
2246:
2240:
2237:
2231:
2228:
2225:
2219:
2199:
2172:
2169:
2166:
2146:
2143:
2140:
2137:
2118:
2115:
2096:
2093:
2085:
2084:
2075:
2044:
2028:
2021:
2015:
2009:
1999:
1986:
1980:
1974:
1971:
1965:
1959:
1939:
1936:
1933:
1913:
1905:
1885:
1865:
1859:
1853:
1850:
1844:
1838:
1818:
1815:
1812:
1792:
1778:
1764:
1761:
1758:
1746:
1726:
1714:
1711:
1705:
1699:
1696:
1687:
1684:
1678:
1675:
1669:
1663:
1644:
1641:
1633:
1614:
1594:
1554:
1534:
1531:
1528:
1516:
1502:
1488:
1474:
1472:
1471:inner regular
1456:
1436:
1416:
1396:
1390:
1384:
1372:
1369:
1361:
1357:
1346:
1342:
1338:
1319:
1316:
1313:
1303:
1289:
1285:
1282:If for every
1269:
1266:
1259:
1243:
1222:
1218:
1214:
1200:
1180:
1172:
1168:
1161:
1151:
1148:
1145:
1141:
1137:
1133:
1127:
1123:
1112:
1109:
1106:
1102:
1097:
1093:
1073:
1045:
1042:
1037:
1033:
1029:
1026:
1023:
1018:
1014:
1010:
1005:
1001:
993:
989:
960:Suppose that
953:
940:
911:
907:
903:
900:
897:
892:
888:
884:
879:
875:
853:
848:
844:
840:
836:
831:
826:
823:
820:
816:
812:
808:
802:
798:
792:
787:
784:
781:
777:
772:
768:
760:
755:
738:
712:
689:
669:
663:
657:
654:
648:
642:
639:
633:
630:
627:
621:
600:
584:
583:disjoint sets
568:
548:
540:
532:
512:
504:
482:
476:
440:
436:
420:
406:
404:
400:
399:
393:
391:
387:
383:
379:
374:
361:
353:
349:
342:
332:
329:
326:
322:
318:
314:
308:
304:
293:
290:
287:
283:
278:
274:
266:
262:
258:
254:
250:
246:
242:
226:
220:
214:
211:
205:
199:
196:
190:
187:
184:
178:
170:
166:
150:
143:
139:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3884:
3877:
3869:
3864:
3842:
3841:
3831:τ-additivity
3787:Additive map
3759:vector space
3748:
3565:
3457:Banach limit
3431:denotes the
3294:real numbers
3271:
3261:
3257:
3253:
3205:
3203:
3192:
3020:
2921:real numbers
2894:
2749:
2740:modular form
2737:
2731:
2725:
2722:
2440:
2308:
2299:
2086:if whenever
2077:
2069:
2068:is called a
2020:set function
2017:
1899:
1784:
1781:Monotonicity
1631:
1508:
1480:
1409:we say that
1228:
959:
756:
534:
526:
435:set function
412:
396:
394:
375:
260:
256:
252:
249:set function
244:
240:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
1219:𝜎-additive
1193:holds then
134:mathematics
3918:Categories
3856:References
3845:PlanetMath
2302:modularity
2012:See also:
2006:See also:
2002:Modularity
1477:Properties
1256:we have a
525:is called
403:modularity
110:April 2024
80:newspapers
3713:μ
3693:μ
3673:μ
3649:…
3537:μ
3517:∞
3485:≤
3473:μ
3470:≤
3419:λ
3378:∩
3369:λ
3366:⋅
3351:∞
3348:→
3325:μ
3280:μ
3216:∅
3162:μ
3157:∞
3142:∑
3118:∞
3103:⋃
3094:μ
3074:…
3055:…
2994:∉
2969:∈
2931:μ
2917:power set
2903:μ
2864:μ
2861:−
2849:μ
2837:∖
2828:μ
2799:μ
2796:−
2784:μ
2761:⊆
2700:∩
2691:μ
2676:∖
2667:μ
2655:∖
2646:μ
2617:∖
2608:∪
2599:∖
2590:∪
2581:∩
2566:∪
2537:∖
2528:∪
2519:∩
2481:∖
2472:∪
2463:∩
2416:μ
2401:μ
2389:∩
2380:μ
2368:∪
2359:μ
2277:ϕ
2262:ϕ
2250:∩
2241:ϕ
2229:∪
2220:ϕ
2170:∩
2141:∪
2081:valuation
2029:μ
1975:μ
1972:≥
1960:μ
1937:⊆
1914:μ
1886:μ
1878:That is,
1854:μ
1851:≤
1839:μ
1816:⊆
1793:μ
1759:≠
1753:∅
1747:μ
1721:∅
1715:μ
1700:μ
1691:∅
1688:∪
1679:μ
1664:μ
1618:∞
1615:−
1595:μ
1575:∞
1555:μ
1523:∅
1517:μ
1489:μ
1457:μ
1437:τ
1417:μ
1385:μ
1373:∈
1347:⋃
1339:μ
1317:τ
1314:∩
1304:⊆
1288:open sets
1267:τ
1201:μ
1162:μ
1157:∞
1142:∑
1118:∞
1103:⋃
1094:μ
1046:…
1027:…
901:…
837:μ
817:∑
778:⋃
769:μ
742:∞
739:−
736:∞
716:∞
693:∞
690:−
658:μ
643:μ
631:∪
622:μ
513:μ
501:(see the
486:∞
480:∞
477:−
421:μ
395:The term
343:μ
338:∞
323:∑
299:∞
284:⋃
275:μ
215:μ
200:μ
188:∪
179:μ
151:μ
3904:21196971
3836:ba space
3781:See also
3250:ba space
2891:Examples
2443:: write
1607:assigns
1567:assigns
1284:directed
1258:topology
992:sequence
867:for any
529:additive
378:measures
257:infinite
169:disjoint
142:function
3500:and if
3258:bounded
3254:bounded
3195:measure
3190:holds.
2919:of the
1509:Either
405:below.
167:of two
94:scholar
3902:
3892:
3751:monoid
3665:, and
3411:where
3206:charge
2309:Given
2076:and a
390:volume
382:length
96:
89:
82:
75:
67:
3755:group
3529:then
3248:(Cf.
2441:Proof
2212:then
2041:on a
1952:then
1898:is a
1831:then
1632:Proof
986:is a
614:then
433:be a
165:union
140:is a
136:, an
101:JSTOR
87:books
3900:OCLC
3890:ISBN
3617:for
3514:<
3455:the
3435:and
3197:and
3193:See
2776:and
2729:and
2555:and
2499:and
2329:and
2159:and
705:and
581:are
561:and
413:Let
386:area
73:news
3508:sup
3443:lim
3341:lim
3264:.)
3228:to
3021:If
2750:If
1785:If
1739:If
1547:or
1469:is
1429:is
1366:sup
1217:or
585:in
533:or
132:In
56:by
3920::
3898:.
3777:.
3552:0.
3236:0.
3204:A
2742:.
2018:A
388:,
384:,
3906:.
3851:.
3729:)
3724:n
3720:A
3716:(
3646:,
3643:2
3640:,
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3625:n
3605:)
3602:1
3599:+
3596:n
3593:,
3590:n
3587:[
3584:=
3579:n
3575:A
3549:=
3546:)
3543:A
3540:(
3511:A
3488:1
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3479:A
3476:(
3467:0
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3393:)
3390:k
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3384:0
3381:(
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3372:(
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3358:1
3345:k
3337:=
3334:)
3331:A
3328:(
3304:R
3262:R
3178:)
3173:n
3169:A
3165:(
3152:1
3149:=
3146:n
3138:=
3134:)
3128:n
3124:A
3113:1
3110:=
3107:n
3098:(
3071:,
3066:n
3062:A
3058:,
3052:,
3047:2
3043:A
3039:,
3034:1
3030:A
3000:.
2997:A
2991:0
2979:0
2972:A
2966:0
2954:1
2948:{
2943:=
2940:)
2937:A
2934:(
2876:.
2873:)
2870:A
2867:(
2858:)
2855:B
2852:(
2846:=
2843:)
2840:A
2834:B
2831:(
2808:)
2805:A
2802:(
2793:)
2790:B
2787:(
2764:B
2758:A
2709:.
2706:)
2703:B
2697:A
2694:(
2688:2
2685:+
2682:)
2679:A
2673:B
2670:(
2664:+
2661:)
2658:B
2652:A
2649:(
2626:,
2623:)
2620:A
2614:B
2611:(
2605:)
2602:B
2596:A
2593:(
2587:)
2584:B
2578:A
2575:(
2572:=
2569:B
2563:A
2543:)
2540:A
2534:B
2531:(
2525:)
2522:B
2516:A
2513:(
2510:=
2507:B
2487:)
2484:B
2478:A
2475:(
2469:)
2466:B
2460:A
2457:(
2454:=
2451:A
2428:.
2425:)
2422:B
2419:(
2413:+
2410:)
2407:A
2404:(
2398:=
2395:)
2392:B
2386:A
2383:(
2377:+
2374:)
2371:B
2365:A
2362:(
2340:,
2337:B
2317:A
2286:)
2283:B
2280:(
2274:+
2271:)
2268:A
2265:(
2259:=
2256:)
2253:B
2247:A
2244:(
2238:+
2235:)
2232:B
2226:A
2223:(
2200:,
2195:S
2173:B
2167:A
2147:,
2144:B
2138:A
2119:,
2116:B
2097:,
2094:A
2054:S
1987:.
1984:)
1981:B
1978:(
1969:)
1966:A
1963:(
1940:B
1934:A
1866:.
1863:)
1860:B
1857:(
1848:)
1845:A
1842:(
1819:B
1813:A
1765:,
1762:0
1756:)
1750:(
1727:.
1724:)
1718:(
1712:+
1709:)
1706:A
1703:(
1697:=
1694:)
1685:A
1682:(
1676:=
1673:)
1670:A
1667:(
1645:,
1642:A
1535:,
1532:0
1529:=
1526:)
1520:(
1397:,
1394:)
1391:G
1388:(
1378:G
1370:G
1362:=
1358:)
1352:G
1343:(
1320:,
1309:A
1299:G
1270:.
1244:,
1239:A
1181:,
1178:)
1173:n
1169:A
1165:(
1152:1
1149:=
1146:n
1138:=
1134:)
1128:n
1124:A
1113:1
1110:=
1107:n
1098:(
1074:,
1069:A
1043:,
1038:n
1034:A
1030:,
1024:,
1019:2
1015:A
1011:,
1006:1
1002:A
971:A
941:.
936:A
912:N
908:A
904:,
898:,
893:2
889:A
885:,
880:1
876:A
854:)
849:n
845:A
841:(
832:N
827:1
824:=
821:n
813:=
809:)
803:n
799:A
793:N
788:1
785:=
782:n
773:(
713:+
670:.
667:)
664:B
661:(
655:+
652:)
649:A
646:(
640:=
637:)
634:B
628:A
625:(
601:,
596:A
569:B
549:A
489:]
483:,
474:[
451:A
362:.
359:)
354:n
350:A
346:(
333:1
330:=
327:n
319:=
315:)
309:n
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294:1
291:=
288:n
279:(
245:k
241:k
227:.
224:)
221:B
218:(
212:+
209:)
206:A
203:(
197:=
194:)
191:B
185:A
182:(
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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