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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.
 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.

## Real Variable.

n the chapter ( Conditional probability chapter ) we introduced notions of random events and illustrated how statements about realization of random variables could be treated as operations on sets and measured via probability function defined on these sets. In the present chapter we expand such approach.

The references for this chapter are [Royden] , [Chung] and [Kolmogorov] .

The triple is called "probability space". Here the is an event space, is a -algebra of subsets of (see the section ( Operations on sets and logical statements )) and is a probability measure (p.m.) , and is -additive. We denote . is the Borel field with and . "Borel field " is a minimal -algebra containing all open sets.

Definition

(Real valued random variable). Let . The mapping is called (" -measurable") "random variable" (r.v.) if for any

Definition

(Borel measurable function) The mapping is a "Borel measurable function" iff

The random variable on induces the triple according to the rule For a measure we introduce a (cumulative) distribution function (d.f.) ,

 A. Operations on sets and logical statements.
 B. Fundamental inequalities.
 C. Function spaces.
 D. Measure theory.
 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.
 Notation. Index. Contents.