Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
a. Complete measure space.
b. Outer measure.
c. Extension of measure from algebra to sigma-algebra.
d. Lebesgue measure.
E. Various types of convergence.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Extension of measure from algebra to sigma-algebra.


(Induced outer measure) Suppose $\QTR{cal}{A}$ is an algebra of sets and MATH has the properties

1. MATH ,

2. MATH for any disjoint collection of sets MATH , MATH , $\forall k$ .

We define the "outer measure induced by $\mu$ " as follows MATH


The $\mu^{\ast}$ introduced in the definition ( Induced outer measure ) is outer measure (see the definition ( Outer measure )).


(Extension of measure to sigma algebra) Under the conditions of the definition ( Induced outer measure ) let $\bar{\mu}$ be the restriction of the outer measure $\mu^{\ast}$ to the algebra $\QTR{cal}{F}$ of $\mu^{\ast}$ -measurable sets. Then

1. MATH is a measure space.

2. if $\mu$ is finite ( $\sigma$ -finite ) then so is $\bar{\mu}$ .

3. if $\mu$ is $\sigma$ -finite then $\bar{\mu}$ is the only extension of $\mu$ to the smallest $\sigma$ -algebra containing $\QTR{cal}{A}$ .

4. if $\mu$ is $\sigma$ -finite then MATH is complete and saturated.

Notation. Index. Contents.

Copyright 2007