INDEX 917

Kolmogorov zero one law, 728Kolmogorov’s inequality, 730Kuratowski theorem, 274

Lagrange multipliers, 216, 217Laplace expansion, 46Laplace transform, 391, 521least squares regression, 203Lebesgue

decomposition, 303Lebesgue integral

desires to be linear, 284nonnegative function, 280other definitions, 283simple function, 281

Lebesgue integral versus Riemann Stieltjesintegral, 295

Lebesgue measurable functionapproximation with Borel measurable,

315Lebesgue measure

approximation with Borel sets, 315one dimensional, 258properties, 315

Lebesgue number, 77, 97Lebesgue points, 324Lebesgue Stieltjes measure, 250, 257, 345left inverse, 38, 39Leray Schauder alternative, 165Levy theorem, 772lim inf, 66

properties, 67lim sup, 66

properties, 67lim sup and lim inf, 288limit

continuity, 185infinite limits, 183point, 71

limit of a function, 183limit of a sequence, 72

well defined, 72limit point, 183, 502limits

combinations of functions, 183existence of limits, 66independent random variables, 729

limits and continuity, 185

Lindeloff property, 76linear

not continuous, 535linear combination, 30, 44, 100linear functional

positive, 318linear independence, 103linear map of measurable set, 330linear maps, 15

closed, 539continuous, 534equivalent conditions, 534

linear relationship, 30linear relationships

row operations, 31linear space, 99linear transformation

defined on a basis, 125dimension of vector space, 125rank m, 220

linear transformationsa vector space, 125sum, 125

linearly dependent, 100linearly independent, 100linearly independent set

enlarging to a basis, 103Lipschitz

continuous, 82functions, 437

Lipschitz functionintegral of its derivative, 437

Lipschitz functions, 394of measurable sets, 329

Lipschitz mapsextension, 442

little o notation, 186local martingale, 864local maximum, 219local minimum, 219local submartingale, 864localizing sequence, 864locally compact, 504locally compact , 504locally finite, 369, 705locally one to one, 233lower semicontinuous, 97Lusin’s theorem, 371