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Halmos P.R. — Measure Theory

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Название: Measure Theory

Автор: Halmos P.R.

Аннотация:

Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1974

Количество страниц: 324

Добавлена в каталог: 30.06.2008

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Предметный указатель

$\mu *$ -measurable sets      44
$\mu *$ -partition      48
$\mu$ -partition      31
$\sigma$ -algebra      28
$\sigma$ -bounded      4
$\sigma$ -compact      4
$\sigma$ -finite measure      31
$\sigma$ -finite measure, which is infinite on every interval      183
$\sigma$ -ring      24
Absolute continuity      124
Absolute continuity for functions of a real variable      181
Absolute continuity for set functions      97
Absolutely normal numbers      206
Additive set functions      30
Additive set functions on semirings      31
Algebra of sets      21
Almost everywhere      86
Almost uniform convergence      89
Approximation of sets in a $\sigma$ -ring by sets in a ring      56
Associated measure ring      167
Associated metric space      168
atom      168
Average theorem      261
Baire, contraction of a Borel measure      229
Baire, functions      223
Baire, measurable functions on a locally compact space      220
Baire, measure      223
Baire, sets      220
Base      3
Base at e      7
Bayes’ Theorem      195
Bernoulli’s theorem      201
Boolean $\sigma$ -algebra      166
Boolean $\sigma$ -ring      166
Boolean algebra      166
Boolean algebra of sets      21
Boolean ring      22 165
Boolean ring of sets      19
Borel — Cantelli lemma      201
Borel, measurable functions on a locally compact space      219
Borel, measurable functions on the real line      78
Borel, measure      223
Borel, sets of a locally compact space      219
Borel, sets of n-dimensional Euclidean space      153
Borel, sets of the real line      62
Bounded convergence theorem      110
Bounded linear functional on $\mathcal{L}$       249
Bounded linear functional on $\mathcal{L}_2$       178
Bounded sets in locally compact spaces      4
Bounded sets in topological groups      7
Bounded variation      123
Cantor, function      83
Cantor, set      67
Cartesian product of measurable spaces      140
Cartesian product of measure spaces      145 152 157
Cartesian product of non $\sigma$ -finite measure spaces      145
Cartesian product of sets      137 150
Cartesian product of topological spaces      4
Cavalieri’s principle      149
Center      5
Characteristic function      15
class      10
Closed sets      3
Closure      3
Coefficient of correlation      196
Collection      10
Compact sets      4
Complement      16
Complete Boolean ring      169
Complete measure      31
Completely regular spaces      5
Completion of a measure      55
Completion of a topological group      8
Completion, regular measure      230
Complex measure      120
Conditional expectation      209
Conditional probability      195 207
Conditional probability as a measure      210
Content      231
Continuity and additivity of infinite valued set functions      40
Continuity and additivity of set functions on rings      39
Continuity and additivity of set functions on semirings      40
Continuity from above and below      39
Continuous transformations      5
Convergence in measure      91
Convergence in the mean      103
Convergence of a series of sets      19
Convergence of sequences of measures      170
Convergence, a.e.      86
Convergence, a.e. and in measure      89 90
Convex metric spaces      169
Convolution      269
Coset      6
Countably, additive      30
Countably, subadditive      41
Cylinder      29 155
Decreasing of sets      16
Decreasing sequences of partitions      171
Dense, sequences of partitions      171
Dense, sets      3
Density theorem      268
Derivatives of set functions      133
Difference of two sets      17
Difference, set      68
DIMENSION      152
Discontinuity from above of regular outer measures      53
Discrete topological space      3
Disjoint      15
Disjoint, sequences of sets      38
Distance between integrable functions      98
Distribution function      80
Domain      161
Double integral      146
Egoroff’s theorem      88
Egoroff’s theorem on a set of infinite measure      90
Elementary function      86
Empty set      10
Entire space      9
Equal sets      10
Equicontinuous      108
Equivalent sequences of functions      201
Equivalent signed measures      126
Essential supremum      86
Essentially bounded functions      86
Exhaustion      76
Extended real number      2
Extension of measures      54
Extension of measures to larger $\sigma$ -rings      71
Fatou’s Lemma      113
Finite and totally finite measure spaces      73
Finite intersection property      4
Finite measure      31
Finitely additive      30
Finitely subadditive      41
Fubini’s Theorem      148
Full sets      52 132
Full subgroups      250
Fundamental in measure      91
Fundamental in the mean      99
Fundamental sequence      87
Generated $\sigma$ -ring      24
Generated hereditary class      41
Generated invariant $\sigma$ -ring      283
Generated monotone class      27
Generated ring      22
Graph      143
Group      6
Haar measure      251
Hahn decomposition      121
Hamel basis      277

Hausdorff measure      53
Hausdorff space      4
Hereditary class      41
Holder’s Inequality      175
Homeomorphism      5
Homomorphism      6
Horizontal line      131
Identity      6
Image      161
Increasing sequence of sets      16
Indefinite integral      97
Independent functions      192
Independent sets      191
Induced Borel measure      234
Induced inner content      232
Induced measure      47
Induced outer measure      42 233
Inequivalence of two definitions of absolute continuity      128
inf      12
Inferior limit      16
Infimum      1
Inner content      232
Inner measure      58
Inner regular content      239
Inner regular set      224
Integrable functions      102
Integrable simple functions      95
integral      95 102
Integration by parts      269
Interior      3
Intersection      13
INTO      161
Invariant $\sigma$ -rings      283
Invariant sets      29
Invariant subgroups      6
Inverse      6
Inverse image      76 161
Isomorphism      167
Iterated integral      146
J-cylinder      155
Jacobian      164
Join      14
Jordan decomposition      123
Kolmogoroff’s inequality      196
Lattice of sets      25
Lebesgue decomposition      134
Lebesgue integral      106
Lebesgue measurable function      78
Lebesgue measurable set      62
Lebesgue measure      62 153
Lebesgue — Stieltjes measure      67
Left Haar measure      252
Left invariance      252
Left translation      6
Lim inf, limit, lim sup      16
Linear functional on $\mathcal{L}$       243
Linear functional on $\mathcal{L}_2$       178
Linearly independent sets      277
Locally bounded      8
Locally compact      4
Lower, ordinate set      142
Lower, variation      122
Lusin’s theorem      243
Mean convergence      103
Mean fundamental      99
Mean value theorem      114
Measurability preserving transformation      164
Measurable cover      50
Measurable function      77
Measurable function of a measurable function      83
Measurable group      257
Measurable kernel      59
Measurable rectangle      140 154
Measurable set      73
Measurable set which is not a Borel set      67 83
Measurable space      73
Measurable transformation      162
Measure      30
Measure in metric spaces      40
Measure on intervals      35
Measure, algebra      167
Measure, preserving transformation      164
Measure, ring      167
Measure, space      73
Meet      14
Metric outer measures      48
Metric spaces      5
Minkowski’s inequality      176
MODULO      127
Monotone class      27
Monotone class generated by a ring      27
Monotone functions of a real variable      179
Monotone sequences      16
Monotone set functions      37
Multiplication theorem      195
Mutually singular      126
Negative part      82
Negative sets      120
Neighborhood      3
Non atomic      168
Non coincidence of complete $\sigma$ -ring and $\sigma$ -ring of $\mu *$ -measurable sets      58
Non measurable sets      69
Non product measures in product spaces      214
Non regular measures      231
Non regular outer measures      52 53 72
Non term by term integrability      111 112
Non uniqueness of extension      57
Norm      171
Normal class      28
Normal numbers      206
Normalized      171
One to one      161
One to one measurable transformation which is not measurability preserving      165
One-point compactification      4
Onto      161
Open covering      4
Open set      3
Open transformation      5
Outer measure      42
Outer measures on metric spaces      48
Outer regular sets      224
Partition      31 47 171
Point      9
Point at infinity      240
Positive linear functional      243
Positive measure      166
Positive part      82
Positive sets      120
Principle of duality      17
Probability measures and spaces      191
Product measures      145
Product of a sequence of measures      157
Product of partitions      32 48
Product of transformations      161
Projection      6
Proper difference      17
Purely atomic      182
Quotient group      6
Rademacher functions      195
RADIUS      5
Radon — Nikodym theorem      128
Radon — Nikodym theorem, counter examples to generalizations      131
RANGE      161
rectangle      137 150 154
Regular contents      237
Regular measures      224
Regular outer measures      52
Regular sets      224
Relative complement      17
Relative topology      3

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