User:Lord Farin/Long-Term Projects/Schilling

Processing of 'Measures, Integrals and Martingales'

2005: René L. Schilling: Measures, Integrals and Martingales: $\S 2$

2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$

First page it covers: Definition:Set Union.

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)

Covered missing proofs up to $\S 4$. --Lord_Farin 19:18, 23 May 2012 (EDT)

Up to $\S 6$. --Lord_Farin 09:32, 26 May 2012 (EDT)

Theorem statements up to $\S 12$. --Lord_Farin 13:57, 10 July 2012 (UTC)

Up to $\S 14$. --Lord_Farin 06:23, 24 July 2012 (UTC)

Proofs up to $\S 7$. --Lord_Farin 18:33, 4 August 2012 (UTC)

Proofs up to $\S 8$. --Lord_Farin 09:34, 5 August 2012 (UTC)

Missing Proofs (or: check existing proof aligns with Schilling's)

• $9.3$: Integral of Positive Simple Function is Additive, Integral of Positive Simple Function is Increasing
• $9.5$: 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
• All from there on

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
• Most problems of $\S 10$-$\S 14$
• $15.1-4$ on a regularity property of Hölder continuous maps under Lebesgue measure whose extent I can't determine atm. and whose formulation in Schilling is very ad hoc.
• In fact, the rest of $\S 15$ as the necessary basis on multidimensional real analysis is missing.

Other things

• Go through the exercises again at the end to post up more, and provide better linking
• Precede the exercise numbering of earlier chapters with chapter number for similarity with rendition in book (e.g. $\S 5$: Problem $5.1$ in place of $\S 5$: Problem $1$)
• Category:Stieltjes Functions, Category:Simple Functions, Category:Dirac Measures, Category:Lebesgue Measure, maybe Category:Discrete Measures
• Better categorisation (also using the above)