Measure TheoryThis book was planned originally not as a work to be published, but as an excuse to buy a computer, incidentally to give me a chance to organize my own ideas ~n what measure theory every would-be analyst should learn, and to detail my approach to the subject. When it turned out that Springer-Verlag thought that the point of view in the book had general interest and offered to publish it, I was forced to try to write more clearly and search for errors. The search was productive. Readers will observe the stress on the following points. The application of pseudometric spaces. Pseudo metric, rather than metric spaces, are applied to obviate the artificial replacement of functions by equivalence classes, a replacement that makes the use of "almost everywhere" either improper or artificial. The words "function" and "the set on which a function has values at least E" can be taken literally in this book. Pseudometric space properties are applied in many contexts. For example, outer measures are used to pseudometrize classes of sets and the extension of a finite measure from an algebra to a 0" algebra is thereby reduced to finding the closure of a subset of a pseudo metric space. |
Contents
| 1 | |
Applications of coordinate space measures | 7 |
77700 | 26 |
10 | 30 |
12 | 33 |
Measure Spaces | 37 |
8 | 43 |
15 | 49 |
Lattice property of the class of signed measures | 146 |
Absolute continuity and singularity of a signed measure | 147 |
Decomposition of a signed measure relative to a measure | 148 |
A basic preparatory result on singularity | 150 |
Bounded linear functionals on L¹ | 151 |
Sequences of signed measures | 152 |
VitaliHahnSaks theorem continued | 155 |
Measures and Functions of Bounded Variation on R | 157 |
Measurability and sequential convergence | 58 |
11 | 65 |
Integration | 73 |
Uniform integrability test functions | 81 |
Conjugate spaces and Hölders inequality | 88 |
Fourier series | 109 |
Fourier series properties | 110 |
Orthogonalization Erhardt Schmidt procedure | 111 |
Fourier trigonometric series | 112 |
Two trigonometric integrals | 113 |
The FourierPlancherel theorem | 115 |
Ergodic theorems | 117 |
Convergence of Measure Sequences | 123 |
Linear functionals on subsets of CS | 126 |
Generation of positive linear functionals by measures S compact metric | 128 |
CS convergence of sequences in MS S compact metric | 131 |
Metrization of MS to match CS convergence compactness of McS S compact metric | 132 |
Properties of the function μuf from MS in the dy metric into R S compact metric | 133 |
Generation of positive linear functionals on CoS by measures S a locally compact but not compact separable metric space | 135 |
CoS and CooS convergence of sequences in MS S a locally compact but not compact separable metric space | 136 |
Metrization of MS to match CoS convergence compactness of McS S a locally compact but not compact separable metric space ca strictly positive nu... | 137 |
Properties of the function uuf from MS in the doм metric into R S a locally compact but not compact separable metric space | 138 |
Stable CoS convergence of sequences in MS S a locally compact but not compact separable metric space | 139 |
Properties of the function µµf from MS in the dì metric into R S a locally compact but not compact separable metric space | 141 |
Application to analytic and harmonic functions | 142 |
Signed Measures 145 | 144 |
Vitali covering of a set | 158 |
Functions of bounded variation | 160 |
Functions of bounded variation vs signed measures | 163 |
Absolute continuity and singularity of a function of bounded variation | 164 |
The convergence set of a sequence of monotone functions | 165 |
Intervals of uniform convergence of a convergent sequence of monotone functions | 166 |
CoR convergence of a measure sequence | 169 |
Stable CoR convergence of a sequence of probability distributions | 171 |
Application to a stable CoR metrization of MR | 172 |
A ratio limit lemma | 174 |
Application to the boundary limits of harmonic functions | 176 |
Conditional Expectations Martingale Theory | 179 |
Conditional expectation properties | 183 |
Filtrations and adapted families of functions | 187 |
Martingale theory definitions | 188 |
Martingale examples | 189 |
Elementary properties of sub super martingales | 190 |
Optional times | 191 |
Optional time properties | 192 |
The optional sampling theorem | 193 |
The maximal submartingale inequality | 194 |
The submartingale upcrossing inequality | 195 |
Backward martingale convergence | 197 |
Backward supermartingale convergence | 198 |
| 208 | |
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Common terms and phrases
absolutely continuous According to Theorem application Borel measurable Borel sets bounded variation Co(R Co(S compact metric space compact separable metric complement conditional expectations context continuous function Coo(S coordinate functions corresponding countable additivity countable union defined definition dense disjunct union equivalence classes everywhere example finite unions finite valued finitely additive following theorem Fourier function f Hilbert space implies increasing sequence indicator functions inequality infinite sequence intersections Lebesgue measure Let f lim inf lim sup martingale mathematical measurable function measurable sets measure space monotone increasing function mutually independent notation null set orthogonal outer measure probability measure probability space Proof pseudometric space random variables right continuous right semiclosed intervals S₁ satisfied Section separable metric space sequence f sequence of measurable sequence of sets set function signed measure strictly positive subadditive submartingale summands supermartingale Suppose supremum trivial uniform integrability uniformly union of sets vectors