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.

## Outer measure.

et be a measure space (see the definition ( Measurable space )).

Definition

(Outer measure) Let be a mapping defined on all subsets of and valued in . The is called "outer measure" if the following conditions are satisfied:

1.

2. .

3. .

Definition

(Measurable set) A set is called " -measurable" if for any set we have

Proposition

-measurable sets constitute a -algebra. The restriction of to such -algebra is a complete measure.

 Notation. Index. Contents.