## A User's Guide to Measure Theoretic ProbabilityThis book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean. |

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who have not had the luxury of taking a course in measure theory.

### Contents

II | 1 |

V | 3 |

VI | 5 |

VII | 7 |

VIII | 11 |

IX | 13 |

X | 14 |

XI | 17 |

LXXIII | 203 |

LXXIV | 205 |

LXXV | 206 |

LXXVI | 208 |

LXXVII | 211 |

LXXVIII | 213 |

LXXIX | 215 |

LXXX | 217 |

XII | 22 |

XIII | 26 |

XIV | 29 |

XV | 31 |

XVI | 33 |

XVII | 36 |

XVIII | 37 |

XIX | 39 |

XX | 41 |

XXI | 43 |

XXII | 45 |

XXIII | 51 |

XXIV | 53 |

XXV | 58 |

XXVI | 59 |

XXVII | 65 |

XXVIII | 68 |

XXIX | 70 |

XXX | 71 |

XXXI | 75 |

XXXII | 77 |

XXXIII | 80 |

XXXIV | 83 |

XXXV | 88 |

XXXVI | 93 |

XXXVII | 95 |

XXXVIII | 97 |

XXXIX | 99 |

XL | 102 |

XLI | 108 |

XLII | 111 |

XLIII | 113 |

XLIV | 116 |

XLV | 118 |

XLVI | 121 |

XLVII | 123 |

XLVIII | 128 |

XLIX | 131 |

L | 135 |

LI | 138 |

LII | 142 |

LIII | 147 |

LIV | 151 |

LV | 152 |

LVI | 153 |

LVII | 155 |

LVIII | 159 |

LIX | 162 |

LX | 166 |

LXI | 169 |

LXII | 176 |

LXIII | 182 |

LXIV | 184 |

LXV | 186 |

LXVI | 190 |

LXVII | 193 |

LXVIII | 195 |

LXIX | 198 |

LXXI | 200 |

LXXII | 202 |

LXXXI | 219 |

LXXXII | 222 |

LXXXIII | 226 |

LXXXIV | 228 |

LXXXV | 230 |

LXXXVI | 234 |

LXXXVII | 237 |

LXXXVIII | 239 |

LXXXIX | 242 |

XC | 244 |

XCI | 248 |

XCII | 249 |

XCIII | 256 |

XCIV | 258 |

XCV | 261 |

XCVI | 264 |

XCVII | 266 |

XCVIII | 268 |

XCIX | 271 |

C | 272 |

CI | 274 |

CII | 275 |

CIII | 276 |

CIV | 278 |

CV | 280 |

CVI | 285 |

CVII | 287 |

CVIII | 289 |

CIX | 291 |

CX | 292 |

CXI | 294 |

CXII | 295 |

CXIII | 296 |

CXIV | 300 |

CXVI | 301 |

CXVII | 302 |

CXVIII | 303 |

CXIX | 305 |

CXX | 306 |

CXXII | 307 |

CXXIII | 308 |

CXXIV | 310 |

CXXV | 312 |

CXXVI | 313 |

CXXVII | 315 |

CXXVIII | 316 |

CXXIX | 317 |

CXXX | 320 |

CXXXI | 324 |

CXXXII | 329 |

CXXXIII | 332 |

CXXXIV | 334 |

CXXXV | 336 |

CXXXVI | 338 |

CXXXVIII | 339 |

CXXXIX | 342 |

CXL | 343 |

CXLI | 345 |

347 | |

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### Common terms and phrases

additive application approximation argument assertion assume assumption Borel bounded Brownian motion called Chapter closed collection compact complete conditional conditional distribution Consider constant construction contains continuous convergence convex corresponding countable Deduce defined definition denote density derivative difference distribution Dominated equals equivalent establish Example exists expectations extends fact filtration finite fixed follows Fourier transform function gives implies increasing independent inequality integrable interval Lebesgue measure Lemma limit linear martingale Mathematical means method metric nonnegative normal Notes Notice numbers positive probability measure Problem proof prove random variables REMARK respect result sample paths Section sequence Show side sigma-field space Statistics stopping subset Suppose surely Theorem theory union vector Write zero