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.
 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.

## Lebesgue measure.

he technique of the proposition ( Extension of measure to sigma algebra ) applied to and the notion of volume yields the following result.

Proposition

(Existence of Lebesgue measure). There exists a -algebra of subsets of and a mapping (Lebesgue measure) such that

1. The includes all open subsets of .

2. If is a cube in then is the volume of .

3. The is -additive: for any collection of subsets such that

4. is a complete measure space.

Remark

Restriction of the above Lebesgue measure to Borel sets is not complete. The Lebesgue measure is complete on the algebra obtained from the algebra of Borel sets by adding all sets included in sets of measure zero (see the proposition ( Completion of measure via addition of null sets )-3).

 Notation. Index. Contents.