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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.

## Measure theory.

efinition

( -algebra) A collection of subsets of a set is called a -algebra if the following conditions hold.

1. .

2. .

3. If , then and .

Definition

(Measurable space) The pair is called a "measurable space" if is a set and is a -algebra of subsets of . A set is called "measurable" if . A mapping is called "measure" if and for a countable collection of disjoint sets. The triple is called "measure space". A set is "locally measurable" if for such that . If, for a measure space , we have then is called "probability space".

Definition

( -finite set). A set is called " -finite" if it is a countable union of measurable sets of finite measure.

 a. Complete measure space.
 b. Outer measure.
 c. Extension of measure from algebra to sigma-algebra.
 d. Lebesgue measure.
 Notation. Index. Contents.