Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.

## Uniform convergence and convergence almost surely. Egorov's theorem.

he concept of almost sure convergence was defined by the formula ( Almost sure convergence ). Equivalently, we characterize the set where the random variables converges to the random variable by the relationship

 (Almost sure convergence 2)

Definition

The sequence converges to uniformly on a set if for s.t. for all .

The key difference from the almost sure convergence is the independence from .

We use the notations for uniform convergence. Using the technique of the section ( Operations on sets and logical statements ) we state that the set satisfies

 (Uniform convergence)
for some function .

Uniform convergence implies the convergence a.s. The inverse relationship is given by the following Egorov's theorem.

Theorem

(Egorov theorem). Let a.s. on then for any small there exists a set s.t. and on .

To prove the Egorov's theorem we need the continuity property of the -additive measure. It serves as a bridge between the convergence concepts defined in terms of set algebra and the topology introduced by the probability measure.

Lemma

(Continuity lemma). If is a -additive probability measure and is an increasing ( ) infinite countable collection of sets then

Proof

Introduce the collection of sets according to . The are disjoint and , . It remains to note that and , hence, .

Corollary

If is increasing then . If is decreasing then .

We now proceed with the prove of the Egorov's theorem.

Proof

The a.s. convergence on implies (see the section ( Operations on sets and logical statements )).

Fix some small as required by the conclusion of the theorem. Our goal is to construct the set as in the formula ( Uniform convergence ) with the property .

Pick some positive integer . We have because restricting from the intersection in all to makes the set bigger. Hence, Consequently, by the lemma ( Continuity lemma ) Note that the set collection increases as increases, hence the set with is the biggest set in the -union. Therefore, Choose so that and form the set By the formula ( Uniform convergence ) we have on . Also, using the formula ( Intersection property )

 Notation. Index. Contents.