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 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.
 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.

## Operations on sets and logical statements.

e start from an example. We would like to describe the notion of almost sure (=almost everywhere) convergence of the random variables to the random variable on the set with respect to a probability . The English statement would be the following: "There exists a subset of the set such that and for any point of the set and any however small positive number there exists an integer depending on and such that for any integer greater then the following inequality holds: ." The mathematical notation for such statements is

 (Almost sure convergence)
The stands for "any", is "there exists some", s.t. is "such that". Every variable is bound by one of the operators or . If the variable is not bound then the logical statement is meaningless.

Next, we describe a situation when does not converge to on Such statement is the exact opposite to the statement above. The general recipe is to permute and and replace the key relationship to the opposite: Again, the boundedness of every variable to or is important.

Let us now describe the notion of convergence in terms of sets. We start from the set of all points where converges to and represent it in terms of more elementary sets: Notably, we describe the ("any") using the intersection and describe the ("exists") using the union of the sets.

Let be the total set. We would like to describe the equivalent of negation: We note the following property of set operations

 (Intersection property)
Hence, We use another property of set operations:

 (Union property)
Thus, Note how this result matches the negative statement obtained via the logical operations.

There will be numerous applications of this technique in the present chapter.

 Notation. Index. Contents.