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.

## Extension of measure from algebra to sigma-algebra.

efinition

(Induced outer measure) Suppose is an algebra of sets and has the properties

1. ,

2. for any disjoint collection of sets , , .

We define the "outer measure induced by " as follows

Proposition

The introduced in the definition ( Induced outer measure ) is outer measure (see the definition ( Outer measure )).

Proposition

(Extension of measure to sigma algebra) Under the conditions of the definition ( Induced outer measure ) let be the restriction of the outer measure to the algebra of -measurable sets. Then

1. is a measure space.

2. if is finite ( -finite ) then so is .

3. if is -finite then is the only extension of to the smallest -algebra containing .

4. if is -finite then is complete and saturated.

 Notation. Index. Contents.