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It does not matter whether it is obvious or not. It is not supplied by reference which relates the two concepts. We can base and judge article text only by the published reliable sources. I strongly recommend you to carefully read and understand the policies I mentioned above. Without references you
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It's not important whether the term is essential or not. I mean this is also not for us to decide and one can use the term also for other things than the definition of
Bochner measurability. How about a compromis, that in the article simple function it will be mentioned shortly in one sentence, that
2188:
Can we reconcile or refine the definition of simple function in this article with the defintion given by the
Springer online encyclopedia of mathematics, which says a simple function is a measureable function that takes on at most a countable number of values?
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means. For the reader it's certainly not obvious that this term is explained in the article
Bochner measurable. And I don't think the article Bochner measurable does a good job because it uses the term before it explains what it is. This is bad
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If the issue is only a source, I can provide one. But for StrokeOfMidnight that is not the issue, the user just doesn't want the concept to be on
Knowledge because "it is obvious". I tried to discuss, but I waited 3 weeks and didn't get an
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does a good-enough job briefly introducing countably valued functions in the intro section, so linking to this article is unhelpful. Linking to the BMF in "See also" section would be more appropriate.
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Go on and delete it, I don't care anymore and have no interest to look for a source anymore, if you talk to me like that. ("..waste other
Wikipedians time.."). I also waste a lot of time.--
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Isn't this at odds with the requirement of finite codomain? I would think of the floor function as being a step function but it takes on infinity values.
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This term is not essential even for the definition of BMF, for which it's intended. Mentioning "countably-valued" just before the formula
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It would be nice to include a section on the standard representation of simple measurable functions. If I have time I'll get to it.
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can be skipped without affecting clarity. Therefore the term is not important enough to be discussed in the current article.
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So, we let n approach infinity. Then our estimate improves dramatically. In fact, as n approaches
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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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The opening paragraph of this article is what the rest of math wikipedia should aspire to.
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segment of the partition we are creating on the range (y) of the function f. Since the
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I think the analysis book by
Kolmogorov and Fomin uses the same definition as Springer
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263:{\displaystyle I_{n,k}=\left[{\frac {k-1}{2^{n}}},{\frac {k}{2^{n}}}\right)}
1645:{\displaystyle [(k-1)/2^{n},(2k-1)/2^{n+1})\cup [(2k-1)/2^{n+1},k/2^{n})}
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segment has to cover the entire remaining range, we extend it from
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simple functions may be generalized to have countable ranges. The
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seems odd, but everything will work out when we take the limit.
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I still want the term too be displayed. It's an important term--
2334:? I am not sure I understand, why you are so much against it.--
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http://www.encyclopediaofmath.org/index.php/Lebesgue_integral
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the function extended to countably values is called
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830:{\displaystyle {\frac {k-1}{2^{n}}},k=2^{2n}+1)}
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2231:article on Bochner measurable functions
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1242:{\displaystyle [(k-1)/2^{n},k/2^{n}),}
402:Someone should explain how it is that
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317:{\displaystyle k=1,2,\ldots ,2^{2n}}
95:This article is within the scope of
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38:It is of interest to the following
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1679:{\displaystyle {\frac {1}{2^{n}}}}
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2217:Countably valued functions - why?
115:Knowledge:WikiProject Mathematics
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476:. I don't think it is correct.
118:Template:WikiProject Mathematics
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2259:but rather about what the term
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469:{\displaystyle I_{n,2^{2n}+1}=}
391:{\displaystyle I_{n,2^{2n}+1}=}
135:This article has been rated as
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1873:{\displaystyle (2k-1)/2^{n+1}}
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2255:the point is, it's not about
2143:all step functions are simple
1406:{\displaystyle )=(k-1)/2^{n}}
109:and see a list of open tasks.
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2307:{\displaystyle f(t)=\ldots }
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2176:21:49, 19 October 2014 (UTC)
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2138:19:17, 4 December 2009 (UTC)
1913:}. Matching the ranges of
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169:19:15, 13 January 2011 (UTC)
1817:{\displaystyle (k-1)/2^{n}}
483:00:12, 8 October 2007 (UTC)
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2355:I didn't get any answer.--
1973:we see that in all cases,
1107:Proof that f is increasing
1062:{\displaystyle 2^{2n}+1th}
679:{\displaystyle 2^{2n}+1th}
637:{\displaystyle 2^{2n}+1th}
2453:01:27, 4 March 2023 (UTC)
2332:countably valued function
2261:countably valued function
2005:{\displaystyle f_{n+1}(x}
1767:{\displaystyle f_{n+1}(x}
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2118:{\displaystyle f_{n}(x)}
837:. I know approximating
546:11:06, 1 June 2008 (UTC)
141:project's priority scale
2060:{\displaystyle f_{n}(x}
1966:{\displaystyle f_{n+1}}
1906:{\displaystyle 2^{n+1}}
1474:{\displaystyle )=2^{n}}
1439:{\displaystyle f_{n}(x}
1351:{\displaystyle f_{n}(x}
1275:{\displaystyle 2^{2n},}
1082:{\displaystyle \infty }
982:{\displaystyle 1/2^{n}}
920:{\displaystyle \infty }
757:{\displaystyle \infty }
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1158:{\displaystyle )=y}
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1027:) Notice that our
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1460:
1457:
1446:
1435:
1432:
1427:
1423:
1400:
1396:
1391:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1358:
1347:
1344:
1339:
1335:
1314:
1311:
1308:
1303:
1299:
1295:
1283:
1271:
1266:
1263:
1259:
1238:
1235:
1230:
1226:
1221:
1217:
1214:
1209:
1205:
1200:
1196:
1193:
1190:
1187:
1184:
1181:
1169:
1165:
1154:
1151:
1148:
1137:
1126:
1123:
1120:
1108:
1105:
1094:129.71.204.146
1078:
1058:
1055:
1052:
1049:
1044:
1041:
1037:
1024:
1013:
1010:
1007:
995:
991:
976:
972:
967:
963:
941:
937:
916:
906:
890:
886:
865:
862:
859:
854:
850:
846:
826:
823:
820:
815:
812:
808:
804:
801:
798:
791:
787:
782:
779:
776:
753:
731:
727:
723:
718:
714:
709:
703:
700:
696:
675:
672:
669:
666:
661:
658:
654:
633:
630:
627:
624:
619:
616:
612:
589:
586:
581:
578:
574:
570:
567:
563:
549:
548:
528:
525:
520:
517:
513:
509:
506:
502:
465:
462:
459:
454:
450:
446:
443:
438:
435:
430:
427:
423:
419:
416:
412:
387:
384:
381:
376:
372:
368:
365:
360:
357:
352:
349:
345:
341:
338:
334:
311:
308:
304:
300:
297:
294:
291:
288:
285:
282:
279:
258:
250:
246:
242:
237:
230:
226:
221:
218:
215:
208:
204:
199:
196:
193:
189:
175:
172:
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:
2487:
2476:
2473:
2471:
2468:
2467:
2465:
2454:
2450:
2446:
2445:Tensorproduct
2442:
2438:
2434:
2430:
2429:Tensorproduct
2426:
2425:
2424:
2420:
2416:
2411:
2410:
2409:
2405:
2401:
2400:Tensorproduct
2396:
2395:
2394:
2393:
2389:
2385:
2381:
2380:
2376:
2366:
2362:
2358:
2357:Tensorproduct
2352:
2347:
2346:
2345:
2341:
2337:
2336:Tensorproduct
2333:
2328:
2327:
2326:
2325:
2321:
2317:
2301:
2298:
2292:
2286:
2275:
2271:
2267:
2266:Tensorproduct
2262:
2258:
2252:
2247:
2246:
2245:
2244:
2240:
2236:
2232:
2228:
2224:
2216:
2214:
2212:
2208:
2202:
2198:
2194:
2192:
2183:
2181:
2178:
2177:
2173:
2169:
2162:Great writing
2161:
2159:
2158:
2154:
2150:
2142:
2140:
2139:
2135:
2131:
2126:
2109:
2101:
2097:
2089:. Therefore,
2054:
2046:
2042:
1999:
1991:
1988:
1985:
1981:
1958:
1955:
1952:
1948:
1925:
1921:
1898:
1895:
1892:
1888:
1865:
1862:
1859:
1855:
1850:
1843:
1840:
1837:
1834:
1809:
1805:
1800:
1793:
1790:
1787:
1761:
1753:
1750:
1747:
1743:
1716:
1711:
1708:
1705:
1701:
1686:th--or else y
1669:
1665:
1661:
1634:
1630:
1625:
1621:
1618:
1613:
1610:
1607:
1603:
1598:
1591:
1588:
1585:
1582:
1573:
1565:
1562:
1559:
1555:
1550:
1543:
1540:
1537:
1534:
1528:
1523:
1519:
1514:
1507:
1504:
1501:
1482:
1466:
1462:
1458:
1433:
1425:
1421:
1398:
1394:
1389:
1382:
1379:
1376:
1370:
1345:
1337:
1333:
1325:. Therefore
1306:
1301:
1297:
1269:
1264:
1261:
1257:
1236:
1228:
1224:
1219:
1215:
1212:
1207:
1203:
1198:
1191:
1188:
1185:
1152:
1149:
1124:
1118:
1106:
1104:
1103:
1099:
1095:
1090:
1056:
1053:
1050:
1047:
1042:
1039:
1035:
1011:
1008:
1005:
974:
970:
965:
961:
939:
935:
904:
888:
884:
857:
852:
848:
821:
818:
813:
810:
806:
802:
799:
796:
789:
785:
780:
777:
774:
729:
725:
721:
716:
712:
707:
701:
698:
694:
673:
670:
667:
664:
659:
656:
652:
631:
628:
625:
622:
617:
614:
610:
587:
584:
579:
576:
572:
568:
565:
561:
552:
547:
544:
526:
523:
518:
515:
511:
507:
504:
500:
491:
487:
486:
485:
484:
481:
477:
457:
452:
448:
441:
436:
433:
428:
425:
421:
417:
414:
410:
399:
379:
374:
370:
363:
358:
355:
350:
347:
343:
339:
336:
332:
309:
306:
302:
298:
295:
292:
289:
286:
283:
280:
277:
256:
248:
244:
240:
235:
228:
224:
219:
216:
213:
206:
202:
197:
194:
191:
187:
173:
171:
170:
167:
163:
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:
2373:
2371:
2331:
2278:
2260:
2256:
2226:
2220:
2205:— Preceding
2195:
2187:
2179:
2165:
2146:
2127:
1483:
1110:
1091:
905:
553:
550:
489:
478:
401:
178:
164:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
2168:Crasshopper
2149:Crasshopper
994:) – f_{n}(x
488:That's the
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2464:Categories
1484:For n+1, y
543:Algebraist
490:definition
166:Goldencako
2398:answer.--
2227:of course
2221:Based on
1282:or else y
1168:. Then y
480:Timhoooey
2264:style.--
2379:WP:CITE
2223:WP:ONUS
2207:undated
2197:Tashiro
764:(viz.
602:is the
179:We set
139:on the
1735:. So
1690:is in
1488:is in
1286:is in
1172:is in
36:scale.
2375:WP:RS
989:: -->
2449:talk
2433:talk
2419:talk
2404:talk
2388:talk
2361:talk
2340:talk
2320:talk
2270:talk
2239:talk
2201:talk
2172:talk
2153:talk
2134:talk
1940:and
1111:Let
1098:talk
990:|f(x
324:and
270:for
2203:)
1880:,
1413:or
876:by
744:to
492:of
131:Low
2466::
2451:)
2435:)
2421:)
2406:)
2390:)
2363:)
2342:)
2322:)
2302:…
2272:)
2241:)
2174:)
2155:)
2136:)
2128:--
1841:−
1824:,
1791:−
1720:∞
1589:−
1574:∪
1541:−
1505:−
1481:.
1380:−
1310:∞
1189:−
1100:)
1092:--
1089:.
1077:∞
915:∞
861:∞
778:−
752:∞
461:∞
398:.
383:∞
296:…
217:−
2447:(
2431:(
2417:(
2402:(
2386:(
2377:/
2359:(
2353::
2349:@
2338:(
2318:(
2299:=
2296:)
2293:t
2290:(
2287:f
2268:(
2253::
2249:@
2237:(
2199:(
2170:(
2151:(
2132:(
2113:)
2110:x
2107:(
2102:n
2098:f
2077:)
2068:0
2055:x
2052:(
2047:n
2043:f
2034:≥
2022:)
2013:0
2000:x
1997:(
1992:1
1989:+
1986:n
1982:f
1959:1
1956:+
1953:n
1949:f
1926:n
1922:f
1899:1
1896:+
1893:n
1889:2
1866:1
1863:+
1860:n
1856:2
1851:/
1847:)
1844:1
1838:k
1835:2
1832:(
1810:n
1806:2
1801:/
1797:)
1794:1
1788:k
1785:(
1775:0
1762:x
1759:(
1754:1
1751:+
1748:n
1744:f
1723:]
1717:,
1712:1
1709:+
1706:n
1702:2
1698:[
1688:0
1670:n
1666:2
1662:1
1640:)
1635:n
1631:2
1626:/
1622:k
1619:,
1614:1
1611:+
1608:n
1604:2
1599:/
1595:)
1592:1
1586:k
1583:2
1580:(
1577:[
1571:)
1566:1
1563:+
1560:n
1556:2
1551:/
1547:)
1544:1
1538:k
1535:2
1532:(
1529:,
1524:n
1520:2
1515:/
1511:)
1508:1
1502:k
1499:(
1496:[
1486:0
1467:n
1463:2
1459:=
1456:)
1447:0
1434:x
1431:(
1426:n
1422:f
1399:n
1395:2
1390:/
1386:)
1383:1
1377:k
1374:(
1371:=
1368:)
1359:0
1346:x
1343:(
1338:n
1334:f
1313:]
1307:,
1302:n
1298:2
1294:[
1284:0
1270:,
1265:n
1262:2
1258:2
1237:,
1234:)
1229:n
1225:2
1220:/
1216:k
1213:,
1208:n
1204:2
1199:/
1195:)
1192:1
1186:k
1183:(
1180:[
1170:0
1166:0
1153:y
1150:=
1147:)
1138:0
1125:x
1122:(
1119:f
1096:(
1057:h
1054:t
1051:1
1048:+
1043:n
1040:2
1036:2
1025:0
1012:x
1009:=
1006:x
996:0
992:0
975:n
971:2
966:/
962:1
940:n
936:f
889:n
885:2
864:]
858:,
853:n
849:2
845:[
825:)
822:1
819:+
814:n
811:2
807:2
803:=
800:k
797:,
790:n
786:2
781:1
775:k
730:n
726:2
722:=
717:n
713:2
708:/
702:n
699:2
695:2
674:h
671:t
668:1
665:+
660:n
657:2
653:2
632:h
629:t
626:1
623:+
618:n
615:2
611:2
588:1
585:+
580:n
577:2
573:2
569:,
566:n
562:I
527:1
524:+
519:n
516:2
512:2
508:,
505:n
501:I
464:]
458:,
453:n
449:2
445:[
442:=
437:1
434:+
429:n
426:2
422:2
418:,
415:n
411:I
386:]
380:,
375:n
371:2
367:[
364:=
359:1
356:+
351:n
348:2
344:2
340:,
337:n
333:I
310:n
307:2
303:2
299:,
293:,
290:2
287:,
284:1
281:=
278:k
257:)
249:n
245:2
241:k
236:,
229:n
225:2
220:1
214:k
207:[
203:=
198:k
195:,
192:n
188:I
143:.
42::
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