918 INDEX

Lyapunov Schmidt procedure, 563

manifoldorientable, 396

mapC1, 225primitive and flips, 225

martingale, 741quadratic variation, 866

martingalesequiintegrable, 851

matrixleft inverse, 49lower triangular, 49right inverse, 49right, left inverse, 49row, column, determinant rank, 50upper triangular, 49

matrixblock multiplication, 17inverse, 29, 37more columns than rows, 35non zero kernel, 35partitioned, 17Schur’s theorem, 19

matrix multiplicationblock, 17

maximal chain, 68maximal estimate

real sub-martingales, 801maximal function

Radon measures, 324McShane’s lemma, 637mean value inequality, 208mean value theorem, 208

Cauchy, 137measurability

limit of simple functions, 239measurable, 245

complex valued, 284equivalent formulations, 238linear combinations, 284multifunction, 271multifunctions, 271

measurable complex functionssimple functions, 288

measurable functions, 237approximation, 240

pointwise limit, 237pointwise limits, 651simple functions, 238

measurable into (−∞,∞], 238measurable representative, 678measurable selection, 272measurable sets, 245measure, 241

inner regular, 253on an algebra, 524outer regular, 253properties, 242vector, 621

measure spacecompletion, 260regular, 362

measurescomplex, 621decreasing sequences of sets, 242increasing sequences of sets, 242regularity, 253, 256tight, 766weak convergence, 770

measures from outer measures, 246metric, 71

properties, 71metric space, 71

compact sets, 78complete, 74completely separable, 75open set, 71separable, 75

min max theorem, 147min-max theorem, 147Minkowski functional, 571Minkowski inequality, 358

integrals, 360Minkowski inequality

backwards, 546Minkowski theorem

for integrals, 373Minkowski’s inequality, 360minor, 47mixed partial derivatives, 198modification, 807mollifier, 366monotone convergence theorem, 282monotone function