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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.

Lebesgue measure.


he technique of the proposition ( Extension of measure to sigma algebra ) applied to $\QTR{cal}{R}^{n}$ and the notion of volume yields the following result.

Proposition

(Existence of Lebesgue measure). There exists a $\sigma$ -algebra $\QTR{cal}{F}$ of subsets of $\QTR{cal}{R}^{n}$ and a mapping (Lebesgue measure) MATH such that

1. The $\QTR{cal}{F}$ includes all open subsets of $\QTR{cal}{R}^{n}$ .

2. If $C$ is a cube in $\QTR{cal}{R}^{n}$ then MATH is the volume of $C$ .

3. The $\mu$ is $\sigma$ -additive: MATH for any collection of subsets MATH such that MATH

4. MATH is a complete measure space.

Remark

Restriction of the above Lebesgue measure to Borel sets is not complete. The Lebesgue measure is complete on the algebra obtained from the algebra of Borel sets by adding all sets included in sets of measure zero (see the proposition ( Completion of measure via addition of null sets )-3).





Notation. Index. Contents.


















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