User:Lord Farin/Long-Term Projects/Schilling

= Processing of 'Measures, Integrals and Martingales' =

$\S 2$

This book I deem useful to develop (mostly) the theory of the function spaces which are paramount examples in Conway's book on functional analysis.

Nice side effect is that measure theory gains another authoritative source.

Errata and solutions to the exercises are available at http://www.motapa.de/measures_integrals_and_martingales/index.html.

Progress thus far
Up to $\S 8$ / p.57 Lord_Farin 15:35, 3 April 2012 (EDT)

Up to $8.9$ / p.62 Lord_Farin 18:10, 4 April 2012 (EDT)

Up to $\S 9$ / p.67 Lord_Farin 09:34, 7 April 2012 (EDT)


 * That's good; over two thirds on the way to proving the fundamental theorem of coincidence of Lebesgue and Riemann integral (under suitable circumstances). --Lord_Farin 09:37, 7 April 2012 (EDT)

Up to $9.7$ / p.70 Lord_Farin 17:53, 7 April 2012 (EDT)

Up to $\S 10$ / p.76 Lord_Farin 09:23, 13 April 2012 (EDT)

Proofs present or added below up to $4.7$ --Lord_Farin 06:30, 26 April 2012 (EDT)

Proofs present or added below up to $\S 5$. $\mathcal A$ replaced by $\Sigma$ up to $\S 5$. --Lord_Farin 06:41, 28 April 2012 (EDT)

As above, up to $\S 6$. --Lord_Farin 13:38, 28 April 2012 (EDT)

As above, up to $\S 7$. --Lord_Farin 10:53, 30 April 2012 (EDT)

$\S 8$. --Lord_Farin 10:46, 12 May 2012 (EDT)

$\S 10$. --Lord_Farin 09:02, 13 May 2012 (EDT)

Missing Proofs

 * $3$ Problem $5(i)$: Generated Sigma-Algebra Preserves Finiteness
 * $3$ Problem $10(ii)$: Borel Sigma-Algebra of Subset is Trace Sigma-Algebra
 * $3$ Problem $11(ii),(iii)$: Generated Sigma-Algebra by Generated Monotone Class
 * $3$ Problem $12$: Borel Sigma-Algebra on Euclidean Space by Monotone Class
 * $4.3 (iv)$: Measure is Strongly Additive
 * $4.9$: Lebesgue Measure Invariant under Translations
 * $4$ Problem $6(ii)$: Series of Measures is Measure
 * $4$ Problem $11(i)$: Lebesgue Measure is Diffuse
 * $4$ Problem $13$: Completion Theorem (Measure Spaces)
 * $5.4$: Dynkin System Closed under Intersections is Sigma-Algebra
 * $5.7$: Uniqueness of Measures
 * $5.8(ii)$: Translation-Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure
 * $6.5$: Lebesgue Pre-Measure is Pre-Measure
 * $6$ Problem $5$: Decomposition of Probability Measures
 * $7.5$: Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings, Characterization of Sigma-Algebra Generated by Collection of Mappings
 * $7$ Problem $7$: Pre-Image Sigma-Algebra of Generated Sigma-Algebra
 * $7$ Problem $9$: Stieltjes Function of Measure is Stieltjes Function, Pre-Measure of Finite Stieltjes Function Extends to Unique Measure, Measure of Stieltjes Function of Measure
 * $7$ Problem $10(iv)$: Cantor Set has Zero Lebesgue Measure
 * $7$ Problem $11$: Factorization Lemma/Real-Valued Function
 * $8.1$: Characterization of Measurable Functions
 * $8.2$: Extended Real Sigma-Algebra Induces Borel Sigma-Algebra on Reals
 * $8.3$: Generators for Extended Real Sigma-Algebra
 * $8.5(ii)$: Simple Function is Measurable
 * $8.7(iii)$: Real-Valued Function is Simple Function iff Finite Image Set
 * $8.7(iv)$: Pointwise Sum of Simple Functions is Simple Function, Pointwise Product of Simple Functions is Simple Function
 * $8.7(v)$: Positive Part of Simple Function is Simple Function, Negative Part of Simple Function is Simple Function
 * $8.7(v)$: Difference of Positive and Negative Parts, Sum of Positive and Negative Parts
 * $8.8$: Measurable Function Pointwise Limit of Simple Functions
 * $8.9$: Pointwise Supremum of Measurable Functions is Measurable, Pointwise Infimum of Measurable Functions is Measurable
 * $8.9$: Pointwise Upper Limit of Measurable Functions is Measurable, Pointwise Lower Limit of Measurable Functions is Measurable
 * $8.9$: Pointwise Limit of Measurable Functions is Measurable
 * $8.10$: Pointwise Sum of Measurable Functions is Measurable, Pointwise Difference of Measurable Functions is Measurable, Pointwise Product of Measurable Functions is Measurable
 * $8.10$: Pointwise Maximum of Measurable Functions is Measurable, Pointwise Minimum of Measurable Functions is Measurable
 * $8.11$: Function Measurable iff Positive and Negative Parts Measurable
 * $8.12$: Measurable Functions Determine Measurable Sets
 * $8.13$: Factorization Lemma/Extended Real-Valued Function
 * $\S 8$: Characterization of Extended Real Sigma-Algebra
 * $8$ Problem $3$: Piecewise Combination of Measurable Mappings is Measurable
 * $8$ Problem $5$: Function Simple iff Positive and Negative Parts Simple
 * $8$ Problem $7$: Bounded Measurable Function Uniform Limit of Simple Functions
 * $9.1$: Integral of Positive Simple Function Well-Defined
 * $9.3(i)$: Integral of Characteristic Function
 * $9.3$: Integral of Positive Simple Function is Positive Homogeneous, Integral of Positive Simple Function is Additive, Integral of Positive Simple Function is Monotone
 * $9.5$: Integral of Positive Measurable Function Extends Integral of Positive Simple Function
 * $9.6$: Beppo Levi's Theorem
 * $9.7$: Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions
 * $9.8$: Integral of Positive Measurable Function is Positive Homogeneous, Integral of Positive Measurable Function is Additive, Integral of Positive Measurable Function is Monotone
 * $9.9$: Series of Positive Measurable Functions is Positive Measurable Function, Integral of Series of Positive Measurable Functions
 * $9.10$: Integral with respect to Dirac Measure, Integral with respect to Discrete Measure
 * $9.11$: Fatou's Lemma for Integrals
 * $9$ Problem $7$: Integral with respect to Series of Measures
 * $9$ Problem $8$: Reverse Fatou's Lemma
 * $9$ Problem $9$: Characteristic Function of Limit Inferior of Sequence of Sets, Characteristic Function of Limit Superior of Sequence of Sets, Fatou's Lemma for Measures
 * $9$ Problem $11$: Kernel Transformation of Measure is Measure, Kernel Transformation of Positive Measurable Function is Positive Measurable Function, Integral with respect to Kernel Transformation of Measure
 * $10.3$: Characterization of Integrable Functions

Skipped thus far (that is, what needs to be done still)

 * Structuring of the definitions colliding with Category:Probability Theory (so far, only $\S 4$)
 * Maybe some more Problems

Other things

 * Category:Sigma-Algebras to be created; maybe also Category:Monotone Classes, Category:Dynkin Systems
 * Go through the exercises again at the end to post up more, and provide better linking
 * Category:Stieltjes Functions, Category:Simple Functions, Category:Dirac Measures, Category:Lebesgue Measure, maybe Category:Discrete Measures