Knowledge (XXG)

Kuratowski–Ulam theorem

Source 📝

427: 177: 99: 244: 271: 468: 129: 356: 492: 320:
with a set of measure zero, comeagre set with one of full measure, and a set with the Baire property with a measurable set.
497: 461: 487: 285:
does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for
502: 395: 454: 305: 72: 379: 28: 182: 44: 309: 352: 338: 249: 124: 106: 438: 404: 344: 48: 366: 304:
The theorem is analogous to the regular Fubini's theorem for the case where the considered
362: 290: 387: 383: 32: 481: 66: 51: 20: 343:. Graduate Texts in Mathematics. Vol. 180. Berlin: Springer. p. 112. 348: 317: 117: 40: 426: 409: 298: 434: 172:{\displaystyle \{x\in X:A_{x}{\text{ is meager (resp. comeager) in }}Y\}} 16:
Analog of Fubini's theorem for arbitrary second countable Baire spaces
313: 403:(1). Institute of Mathematics Polish Academy of Sciences: 247–251. 316:
in a product space, with the usual correspondences, namely,
388:"Quelques propriétés topologiques du produit combinatoire" 442: 252: 185: 132: 75: 65:
be second countable Baire spaces (or, in particular,
265: 238: 171: 93: 36: 462: 8: 224: 218: 166: 133: 469: 455: 408: 257: 251: 203: 190: 184: 160: is meager (resp. comeager) in  158: 152: 131: 74: 329: 101:. Then the following are equivalent if 39:), called also the Fubini theorem for 7: 423: 421: 441:. You can help Knowledge (XXG) by 94:{\displaystyle A\subset X\times Y} 14: 337:Srivastava, Shashi Mohan (1998). 297:a Hausdorff space with countable 425: 239:{\displaystyle A_{x}=\pi _{Y}} 233: 209: 1: 519: 420: 349:10.1007/978-3-642-85473-6 266:{\displaystyle \pi _{Y}} 179:is comeager in X, where 120:(respectively comeager). 29:Kazimierz Kuratowski 410:10.4064/fm-19-1-247-251 396:Fundamenta Mathematicae 310:characteristic function 273:is the projection onto 25:Kuratowski–Ulam theorem 493:Descriptive set theory 340:A Course on Borel Sets 267: 240: 173: 95: 380:Kuratowski, Kazimierz 268: 241: 174: 96: 498:Theorems in topology 250: 183: 130: 73: 263: 236: 169: 91: 43:, is an analog of 450: 449: 161: 510: 488:General topology 471: 464: 457: 435:topology-related 429: 422: 414: 412: 392: 371: 370: 334: 272: 270: 269: 264: 262: 261: 245: 243: 242: 237: 208: 207: 195: 194: 178: 176: 175: 170: 162: 159: 157: 156: 100: 98: 97: 92: 49:second countable 45:Fubini's theorem 27:, introduced by 518: 517: 513: 512: 511: 509: 508: 507: 478: 477: 476: 475: 418: 390: 384:Ulam, Stanislaw 378: 375: 374: 359: 336: 335: 331: 326: 291:Hausdorff space 253: 248: 247: 199: 186: 181: 180: 148: 128: 127: 71: 70: 17: 12: 11: 5: 516: 514: 506: 505: 503:Topology stubs 500: 495: 490: 480: 479: 474: 473: 466: 459: 451: 448: 447: 430: 416: 415: 373: 372: 357: 328: 327: 325: 322: 279: 278: 260: 256: 235: 232: 229: 226: 223: 220: 217: 214: 211: 206: 202: 198: 193: 189: 168: 165: 155: 151: 147: 144: 141: 138: 135: 121: 107:Baire property 90: 87: 84: 81: 78: 47:for arbitrary 33:Stanislaw Ulam 15: 13: 10: 9: 6: 4: 3: 2: 515: 504: 501: 499: 496: 494: 491: 489: 486: 485: 483: 472: 467: 465: 460: 458: 453: 452: 446: 444: 440: 437:article is a 436: 431: 428: 424: 419: 411: 406: 402: 398: 397: 389: 385: 381: 377: 376: 368: 364: 360: 358:0-387-98412-7 354: 350: 346: 342: 341: 333: 330: 323: 321: 319: 315: 311: 307: 302: 300: 296: 292: 289:an arbitrary 288: 284: 276: 258: 254: 230: 227: 221: 215: 212: 204: 200: 196: 191: 187: 163: 153: 149: 145: 142: 139: 136: 126: 122: 119: 115: 112: 111: 110: 108: 104: 88: 85: 82: 79: 76: 68: 67:Polish spaces 64: 60: 55: 53: 50: 46: 42: 38: 34: 31: and 30: 26: 22: 443:expanding it 432: 417: 400: 394: 339: 332: 303: 294: 286: 282: 280: 274: 113: 102: 62: 58: 56: 52:Baire spaces 24: 18: 69:), and let 21:mathematics 482:Categories 324:References 318:meagre set 255:π 228:× 216:∩ 201:π 140:∈ 86:× 80:⊂ 386:(1932). 306:function 281:Even if 246:, where 105:has the 41:category 367:1619545 35: ( 365:  355:  314:subset 299:π-base 118:meager 23:, the 433:This 391:(PDF) 312:of a 308:is a 439:stub 353:ISBN 293:and 123:The 61:and 57:Let 37:1932 405:doi 345:doi 125:set 116:is 19:In 484:: 401:19 399:. 393:. 382:; 363:MR 361:. 351:. 301:. 109:: 54:. 470:e 463:t 456:v 445:. 413:. 407:: 369:. 347:: 295:Y 287:X 283:A 277:. 275:Y 259:Y 234:] 231:Y 225:} 222:x 219:{ 213:A 210:[ 205:Y 197:= 192:x 188:A 167:} 164:Y 154:x 150:A 146:: 143:X 137:x 134:{ 114:A 103:A 89:Y 83:X 77:A 63:Y 59:X

Index

mathematics
Kazimierz Kuratowski
Stanislaw Ulam
1932
category
Fubini's theorem
second countable
Baire spaces
Polish spaces
Baire property
meager
set
Hausdorff space
π-base
function
characteristic function
subset
meagre set
A Course on Borel Sets
doi
10.1007/978-3-642-85473-6
ISBN
0-387-98412-7
MR
1619545
Kuratowski, Kazimierz
Ulam, Stanislaw
"Quelques propriétés topologiques du produit combinatoire"
Fundamenta Mathematicae
doi

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.