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Maximal ergodic theorem

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Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem",
69: 588: 47: 671: 76: 691: 58: 686: 215: 581: 645: 133: 676: 512:, Institute of Mathematical Statistics Lecture Notes - Monograph Series, vol. 48, pp. 248–251, 83: 638: 513: 531: 179: 173: 622: 565: 477:{\displaystyle \int _{f^{*}>\lambda }f\,d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}} 523: 265: 495: 39: 120: 665: 610: 618: 387:{\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.} 561: 124: 21: 527: 553: 116: 518: 15: 148: 626: 569: 43: 406: 298: 268: 218: 182: 136: 476: 386: 281: 254: 200: 164: 313: 646: 589: 494:This theorem is used to prove the point-wise 397:Then the maximal ergodic theorem states that 255:{\displaystyle f\in L^{1}(\mu ,\mathbb {R} )} 8: 471: 452: 48:introducing citations to additional sources 653: 639: 596: 582: 517: 459: 433: 416: 411: 405: 375: 353: 342: 328: 316: 303: 297: 273: 267: 245: 244: 229: 217: 181: 147: 146: 135: 38:Relevant discussion may be found on the 165:{\displaystyle (X,{\mathcal {B}},\mu )} 7: 607: 605: 550: 548: 625:. You can help Knowledge (XXG) by 568:. You can help Knowledge (XXG) by 14: 210:measure-preserving transformation 609: 552: 31:relies largely or entirely on a 20: 249: 235: 208:is a (possibly noninvertible) 192: 159: 137: 1: 682:Theorems in dynamical systems 708: 604: 547: 528:10.1214/074921706000000266 510:Dynamics & Stochastics 59:"Maximal ergodic theorem" 201:{\displaystyle T:X\to X} 113:maximal ergodic theorem 621:-related article is a 564:-related article is a 478: 388: 364: 283: 256: 202: 166: 123:, a discipline within 479: 389: 338: 284: 282:{\displaystyle f^{*}} 257: 203: 167: 672:Probability theorems 404: 296: 266: 216: 180: 134: 44:improve this article 692:Chaos theory stubs 474: 384: 327: 279: 252: 198: 162: 687:Probability stubs 634: 633: 577: 576: 336: 312: 174:probability space 109: 108: 94: 699: 655: 648: 641: 613: 606: 598: 591: 584: 556: 549: 540: 521: 483: 481: 480: 475: 464: 463: 429: 428: 421: 420: 393: 391: 390: 385: 380: 379: 363: 352: 337: 329: 326: 308: 307: 288: 286: 285: 280: 278: 277: 261: 259: 258: 253: 248: 234: 233: 207: 205: 204: 199: 171: 169: 168: 163: 152: 151: 104: 101: 95: 93: 52: 24: 16: 707: 706: 702: 701: 700: 698: 697: 696: 662: 661: 660: 659: 603: 602: 545: 538: 507: 504: 496:ergodic theorem 455: 412: 407: 402: 401: 371: 299: 294: 293: 269: 264: 263: 225: 214: 213: 178: 177: 132: 131: 105: 99: 96: 53: 51: 37: 25: 12: 11: 5: 705: 703: 695: 694: 689: 684: 679: 677:Ergodic theory 674: 664: 663: 658: 657: 650: 643: 635: 632: 631: 614: 601: 600: 593: 586: 578: 575: 574: 557: 543: 542: 536: 503: 500: 485: 484: 473: 470: 467: 462: 458: 454: 451: 448: 445: 442: 439: 436: 432: 427: 424: 419: 415: 410: 395: 394: 383: 378: 374: 370: 367: 362: 359: 356: 351: 348: 345: 341: 335: 332: 325: 322: 319: 315: 311: 306: 302: 276: 272: 251: 247: 243: 240: 237: 232: 228: 224: 221: 197: 194: 191: 188: 185: 161: 158: 155: 150: 145: 142: 139: 121:ergodic theory 107: 106: 42:. Please help 28: 26: 19: 13: 10: 9: 6: 4: 3: 2: 704: 693: 690: 688: 685: 683: 680: 678: 675: 673: 670: 669: 667: 656: 651: 649: 644: 642: 637: 636: 630: 628: 624: 620: 615: 612: 608: 599: 594: 592: 587: 585: 580: 579: 573: 571: 567: 563: 558: 555: 551: 546: 539: 537:0-940600-64-1 533: 529: 525: 520: 515: 511: 506: 505: 501: 499: 497: 492: 490: 468: 465: 460: 456: 449: 446: 443: 440: 437: 434: 430: 425: 422: 417: 413: 408: 400: 399: 398: 381: 376: 372: 368: 365: 360: 357: 354: 349: 346: 343: 339: 333: 330: 323: 320: 317: 309: 304: 300: 292: 291: 290: 274: 270: 241: 238: 230: 226: 222: 219: 211: 195: 189: 186: 183: 175: 156: 153: 143: 140: 130:Suppose that 128: 126: 122: 118: 114: 103: 92: 89: 85: 82: 78: 75: 71: 68: 64: 61: –  60: 56: 55:Find sources: 49: 45: 41: 35: 34: 33:single source 29:This article 27: 23: 18: 17: 627:expanding it 619:chaos theory 616: 570:expanding it 559: 544: 519:math/0004070 509: 493: 488: 487:for any λ ∈ 486: 396: 129: 112: 110: 97: 87: 80: 73: 66: 54: 30: 562:probability 212:, and that 125:mathematics 666:Categories 502:References 100:March 2024 70:newspapers 469:λ 461:∗ 450:μ 447:⋅ 444:λ 441:≥ 438:μ 426:λ 418:∗ 409:∫ 369:∘ 358:− 340:∑ 321:≥ 305:∗ 275:∗ 262:. Define 239:μ 223:∈ 193:→ 157:μ 40:talk page 176:, that 117:theorem 84:scholar 534:  86:  79:  72:  65:  57:  617:This 560:This 514:arXiv 172:is a 115:is a 91:JSTOR 77:books 623:stub 566:stub 532:ISBN 466:> 423:> 111:The 63:news 524:doi 314:sup 289:by 119:in 46:by 668:: 530:, 522:, 498:. 491:. 127:. 654:e 647:t 640:v 629:. 597:e 590:t 583:v 572:. 541:. 526:: 516:: 489:R 472:} 457:f 453:{ 435:d 431:f 414:f 382:. 377:i 373:T 366:f 361:1 355:N 350:0 347:= 344:i 334:N 331:1 324:1 318:N 310:= 301:f 271:f 250:) 246:R 242:, 236:( 231:1 227:L 220:f 196:X 190:X 187:: 184:T 160:) 154:, 149:B 144:, 141:X 138:( 102:) 98:( 88:· 81:· 74:· 67:· 50:. 36:.

Index


single source
talk page
improve this article
introducing citations to additional sources
"Maximal ergodic theorem"
news
newspapers
books
scholar
JSTOR
theorem
ergodic theory
mathematics
probability space
measure-preserving transformation
ergodic theorem
arXiv
math/0004070
doi
10.1214/074921706000000266
ISBN
0-940600-64-1
Stub icon
probability
stub
expanding it
v
t
e

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