Taylor J.C. — An Introduction to Measure and Probability :: Электронная библиотека попечительского совета мехмата МГУ

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Taylor J.C. — An Introduction to Measure and Probability

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Название: An Introduction to Measure and Probability

Автор: Taylor J.C.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 324

Добавлена в каталог: 16.12.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$2\pi$ -periodic integrable function      158
-bounded      238
-martingale convergence theorem      238
$\mathfrak{F}$ -measurable      32
$\mathfrak{F}$ -simple function      30
$\nu$ -negative set      62
$\nu$ -positive set      62
$\sigma$ -additive      9
$\sigma$ -additive probability on a Boolean algebra      17
$\sigma$ -algebra      9
$\sigma$ -algebra generated by $\mathfrak{C}$       14
$\sigma$ -algebra of $\mathbf{P}*$ -measurable sets      24
$\sigma$ -algebra of Borel subsets of $\mathbb{R}$       14
$\sigma$ -algebra of Borel subsets of $\mathbb{R}_2$       87
$\sigma$ -algebra of Borel subsets of $\mathbb{R}_n$       87
$\sigma$ -algebra of Borel subsets of E      131
$\sigma$ -algebra of Lebesgue measurable subsets of $\mathbb{R}_n$       104
$\sigma$ -field      10
$\sigma$ -finite measure      44
$\sigma$ -finite measure space      45
$\sigma$ -finite signed measure      61
Absolutely continuous function on [a,b]      76
Absolutely continuous probability      57
Absolutely continuous with respect to $\mu$       66
Absolutely convergent      54
Absolutely summable      146
Adapted      229
Almost isomorphic measure spaces      206
Analyst’s distribution function      200
Approximate identity      191
Approximation to the identity      191
Archimedean property      2
Arithmetic mean      142
At most countable      15
atoms      224 230
Axiom of the least upper bound      2
Backward martingale      248
Banach algebra      134
Banach space      146
Bessel’s inequality      155
Betting system      232
Bolzano — Weierstrass property      28
Boolean algebra      6
Boolean ring      45
Borel function      36
Borel — Cantelli lemma      148
Borel’s strong law of large numbers      205
Bounded interval      4
Bounded variation      71
Cantor discontinuum      8
Cantor set      8
Cantor — Lebesgue function      84
Cantor’s diagonal argument      15
Cartesian product      15
Cauchy sequence      146
Cauchy — Schwarz inequality      59 216
Central limit theorem: i.i.d. case      275
Central limit theorem: Lindebergh condition      280
Central limit theorem: triangular arrays      281
Cesaro mean      162
Characteristic function      79 263
Characteristic function of a set      30
Chebychev’s Inequality      169
Closed      14
Closed interval      4
Closed monotone class      134
Closure      15
Compact      20
Compact support      150
Compact support in an open set      150
Complete measure space      105
Complete metric space      146
Complete orthonormal system      see Orthonormal system
Completing the measure      see Completion
Completion      47 105
Completion of $\mathfrak{B}(\mathbb{R})$       81
Complex conjugate of a complex number      289
Conditional distribution function of X given Y=y      221
Conditional expectation of X      212
Conditional probability of E given A      47
Conditionally convergent      54
Conditionally independent      226
Conjugate indices      141
Continuity theorem, characteristic function      269
Continuous at $X_o\inE$       36
Continuous at $X_o\in\mathbb{R}$       36
Continuous measure      83
Continuous on $\mathbb{R}$       36
Continuous on E      36
Converge in distribution      255
Converge in law      255
Converge weakly      251 255
Convergence $\mathbf{P}$ -a.e.      167
Convergence $\mu$ -a.e      167
Convergence in measure      168
Convergence in probability      168
Converges to $+\infty$       2
Converges to 0 in       145
Converges to 0 in -norm      145
Converges to B      2
Converges to x      146
Converges uniformly      134
Converges weakly      251
Convex      144
Convex set      214
Convolution kernels      120
Convolution of integrable functions      157
Convolution of two probabilities      108
Countable      15
Countable additivity      9
Countably additive      9
Countably generated      248
Countably subadditive      21
Dedekind cut      2
Dense      16
Dini derivatives      194
Dirac measure at a      26
Dirac measure at the origin      26
Dirichlet’s kernel      162
Discrete distribution function      57
Discrete measure      83
Distribution function      11
Distribution of a random vector X      88
Distribution of X      37
Doob decomposition      243
Doob’s -inequality: countable case      239
Doob’s maximal inequality: finite case      237
Egororov’s theorem      168 202
Empirical distribution function      250
Equicontinuous      283
Ergodic probability space      94
Essential supremum      144
Essentially bounded      144
Excessive function      231
Expectation      30
Expectation of X      38
Exponentially distributed (with parameter $\lambda$ )      56
Fatou’s Lemma      39
Fejer kernel      160
Fejer’s theorem      163
Filtration      225 229
Finite expectation      42
Finite measure      44
Finite measure space      45
Finitely additive probability space      6
Fourier inversion theorem      271
Fourier series      154
Fourier transform      79
Fubini’s theorem for -functions      106
Fubini’s theorem for integrable random variables      100

Fubini’s theorem for Lebesgue integrable functions on $\mathbb{R}^n$       131
Fubini’s theorem for positive
Fubini’s theorem for positive functions      105
g.l.b.      see Greatest lower bound
Gamma distribution      56
Geometric mean      142
Glivenko — Cantelli theorem      261
Greatest lower bound      2
Hahn decomposition      64
Half-open interval      4
Hardy — Littlewood maximal function      187
Harmonic function      231
Heaviside function      26
Heine — Borel theorem      19
Helly’s first selection principle      269
Hermite polynomial      80
Hilbert space      154 216
Holder’s Inequality      142
Homogeneous Markov chain      119
I.i.d. sequence      170
Image measure      138
Image of P      37
Improper Riemann integral of f over $[0,+\infty)$       54
Increasing process      243
Independent collections of events      89
Independent events      89
Independent family of classes of events      132
Independent family of events or sets      111
Independent family of random variables      111
Independent, finite collection of random variables      94
Indicator function of a set      30
Inequality of Cauchy — Schwarz      139
inf      see Infimum
Infimum      20
Infinite product of a sequence of probability spaces      118
Infinite product set      113
Infinite series      3
Infinitely differentiable function      79 see
Initial distribution      122
Inner product      59
Integers      1
Integrable random variable      41 see
integral      30
Integral of X      38
Integration by parts      135
Integration by parts: functions      281
Integration by parts: series      241
interval      4
Inverse of a distribution function      259
Inversion theorem, characteristic function      272
Irrational      1
Isomorphic measure spaces      206
Jensen’s Inequality      144
Jensen’s inequality, conditional expectation      219
Joint distribution, finite-dimensional      110
Jordan decomposition      65
Khintchine’s weak law of large numbers      172
Kolmogorov’s 0-1 law      111
Kolmogorov’s criterion      179
Kolmogorov’s inequality      178 237
Kolmogorov’s strong law of large numbers      177 240
Kolmogorov’s strong law: backward martingale      249
Kronecker’s lemma      241
l.u.b.      see Least upper bound
Laplace transform      79
Law of a random vector X      see Distribution of a random vector X
Law of X      see Distribution of X
Least squares      218
Least upper bound      1
Lebesgue decomposition theorem for measures      68
Lebesgue integral      48
Lebesgue measurable sets      51
Lebesgue measure      46
Lebesgue measure on $\mathbb{R}$       104
Lebesgue measure on $\mathfrak{B}(\mathbb{R})$       104
Lebesgue measure on E      48
Lebesgue point      188
Lebesgue set      188
Lebesgue’s differentiation theorem      188
Left continuous      11
Left limit      11
Liminf      see Limit infimum
Limit infimum      33
Limit supremum      33
Limsup      see limit supremum
Lindebergh condition      280
Lipschitz truncation of a random variable at height c>0      181
Locally integrable function      189
Lower bound      1
Lower semi-continuous      191
Lower semi-continuous at       191
Lower sum      49
Lower variation      65
Lusin’s theorem      202
Marginal distributions      89
Markov chain      119
Markov chain with transition probability $\mathbf{N}$ , initial distribution $\mathbf{P_0}$       127
Markov process      225
Markov process with transition kernel N      225
Markov property      127 226
Markovian kernel      see Transition kernel
Martingale      229
Martingale convergence theorem: countable case      246
Martingale transform      232
Maximal function      see Hardy — Littlewood maximal function
Maximal function: martingale      239
Maximal function: submartingale      237
Measurable      86
Measurable function      32
Measurable space      32
Measure space      44
Measure zero      105
Metric      146
Metric space      146
Metric space E      48
Minkowski’s inequality      142
Modulus of a complex number      289
Monotone class      25
Monotone class of functions      134
Monotone class theorem for functions      133
Monotone class theorem for sets      25
Monotone class theorem, Dynkin’s version      92
Monotone rearrangement      260
Multivariate normal distribution      107
Mutually singular      65
N-dimensional random vector      88
Natural numbers      1
Nearest neighbours      120
Neighbourhood of       36
Non-increasing rearrangement      260
Non-Lebesgue measurable set      49
Non-negative integrable random variable      38
Non-negative measure      44
Norm      58
Normal number      171
Normally summable      146
Normed vector space      146
Null function      41
Null set      104 see
Null variable      41
Occurrence of a set prior to a stopping time      234
open      14
Open ball in a metric space about a of radius $\epsilon$       146
Open interval      4
Open relative to E      see Open subset of E
Open set in a metric space      146
Open subset of $\mathbb{R}^2$       87
Open subset of $\mathbb{R}^n$       87
Open subset of E      131
Optional stopping theorem: finite case      235

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