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.

## Infinitely often events. Borel-Cantelli lemma.

efinition

(Limsup and liminf for sets) ; ;

Proposition

if and only if belongs to infinitely many sets . if and only if it belong to all starting from some .

Proof

We use the technique of the section ( Operations on sets and logical statements ). The statement means exactly The statement means exactly

Definition

. The "i.o." stands for "infinitely often".

Proposition

(Borel-Cantelli lemma, part 1). .

Proof

The sequence of sets is decreasing as . Hence, . Consequently, because and

Proposition

(Borel-Cantelli lemma, part 2). If the sets are independent ( , ) then .

Proof

Note that . Hence, consider By independence, Every term in the above sum is zero because .

Proposition

(IO criteria for AS convergence) For a sequence of r.v. the following are equivalent statements:

1. a.s.

2. ,

Proof

According to the section ( Operations on sets and logical statements ) the a.s. if the following set has the complement of probability zero: Observe that the composition is the " i.o." and since we take the intersection in , the set " i.o." has the complement of probability zero for every . This argument works in either direction.

 Notation. Index. Contents.