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Luzin N property

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302: 150: 106: 71: 170: 343: 372: 336: 289: 329: 207:, as the image of a countable set is countable and thus a null set, but not to functions differentiable on a 203:
Any differentiable function has the Luzin N property. This extends to functions that are differentiable on a
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of that set is zero, then it is measurable and its Lebesgue measure is zero as well.
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does not have the Luzin N property, as the Lebesgue measure of the
317: 35:(also called Luzin property or N property) if for all 158: 114: 79: 41: 219:is zero, but its image is the complete interval. 164: 144: 100: 65: 256:"Luzin-N-property - Encyclopedia of Mathematics" 337: 8: 344: 330: 157: 113: 78: 40: 247: 7: 298: 296: 187:, but since the Lebesgue measure is 191:, it follows that if the Lebesgue 179:Note that the image of such a set 14: 300: 269:Rudin, Real and Complex analysis 145:{\displaystyle \lambda (f(N))=0} 238:and has the Luzin N property. 133: 130: 124: 118: 89: 83: 60: 48: 1: 101:{\displaystyle \lambda (N)=0} 316:. You can help Knowledge by 373:Mathematical analysis stubs 290:Encyclopedia of Mathematics 389: 295: 271:, Lemma 7.25 implies this 66:{\displaystyle N\subset } 27:on the interval has the 165:{\displaystyle \lambda } 312:–related article is a 166: 146: 102: 67: 16:Measure theory concept 310:mathematical analysis 230:if and only if it is 228:absolutely continuous 167: 147: 103: 68: 226:on the interval is 156: 112: 77: 39: 183:is not necessarily 162: 142: 98: 63: 325: 324: 236:bounded variation 380: 346: 339: 332: 304: 297: 285:Luzin-N-property 272: 266: 260: 259: 252: 174:Lebesgue measure 171: 169: 168: 163: 151: 149: 148: 143: 107: 105: 104: 99: 72: 70: 69: 64: 29:Luzin N property 388: 387: 383: 382: 381: 379: 378: 377: 353: 352: 351: 350: 281: 276: 275: 267: 263: 254: 253: 249: 244: 213:Cantor function 205:cocountable set 201: 172:stands for the 154: 153: 110: 109: 108:, there holds: 75: 74: 37: 36: 17: 12: 11: 5: 386: 384: 376: 375: 370: 368:Measure theory 365: 355: 354: 349: 348: 341: 334: 326: 323: 322: 305: 294: 293: 280: 279:External links 277: 274: 273: 261: 246: 245: 243: 240: 200: 197: 161: 141: 138: 135: 132: 129: 126: 123: 120: 117: 97: 94: 91: 88: 85: 82: 62: 59: 56: 53: 50: 47: 44: 31:, named after 15: 13: 10: 9: 6: 4: 3: 2: 385: 374: 371: 369: 366: 364: 363:Real analysis 361: 360: 358: 347: 342: 340: 335: 333: 328: 327: 321: 319: 315: 311: 306: 303: 299: 292: 291: 286: 283: 282: 278: 270: 265: 262: 257: 251: 248: 241: 239: 237: 233: 229: 225: 220: 218: 214: 210: 206: 198: 196: 194: 193:outer measure 190: 186: 182: 177: 175: 159: 139: 136: 127: 121: 115: 95: 92: 86: 80: 57: 54: 51: 45: 42: 34: 33:Nikolai Luzin 30: 26: 23:, a function 22: 318:expanding it 307: 288: 264: 250: 223: 221: 202: 180: 178: 28: 24: 18: 222:A function 21:mathematics 357:Categories 242:References 232:continuous 217:Cantor set 209:conull set 199:Properties 185:measurable 73:such that 160:λ 116:λ 81:λ 46:⊂ 234:, is of 189:complete 152:, where 287:in the 211:: The 308:This 314:stub 19:In 359:: 176:. 345:e 338:t 331:v 320:. 258:. 224:f 181:N 140:0 137:= 134:) 131:) 128:N 125:( 122:f 119:( 96:0 93:= 90:) 87:N 84:( 61:] 58:b 55:, 52:a 49:[ 43:N 25:f

Index

mathematics
Nikolai Luzin
Lebesgue measure
measurable
complete
outer measure
cocountable set
conull set
Cantor function
Cantor set
absolutely continuous
continuous
bounded variation
"Luzin-N-property - Encyclopedia of Mathematics"
Rudin, Real and Complex analysis
Luzin-N-property
Encyclopedia of Mathematics
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Measure theory
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