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.
 a. Uniform convergence and convergence almost surely. Egorov's theorem.
 b. Convergence in probability.
 c. Infinitely often events. Borel-Cantelli lemma.
 d. Integration and convergence.
 e. Convergence in Lp.
 f. Vague convergence. Convergence in distribution.
 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.

## Integration and convergence. efinition

"Simple random variable" (simple function) is a r.v. of the form where the is the indicator function of the event (set) : and are scalars.

Definition

The expectation (integral) of a simple random variable is For any positive random variable we define the integral as where the is taken over all simple positive r.v. such that for .

If the is finite then the r.v. (function of ) is called "summable" on .

Proposition

(Fatou lemma) If is a sequence of non-negative random variables and a.s. on then Proof

It is sufficient to show that for any simple r.v. such that on we have for any small and sufficiently large .

We start by writing definition of a.s. convergence according to the recipes of the section ( Operations on sets and logical statements ) up to a set of measure 0. We fix some then Let's introduce the notation By positiveness of the and we have for any simple function , . The is an increasing sequence, hence, by the proposition ( Continuity lemma ) Pick such that . For we have The last two terms are arbitrarily small and the statement holds for sufficiently large and any simple r.v. .

Proposition

(Dominated convergence theorem) Let is such that a.s. on , and . Then .

Proof

Apply Fatou lemma to and .

 Notation. Index. Contents.