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.

## Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.

et be a probability measure, for a -algebra . An -measurable random variable : gives rise to the set function given by the relationship The measure does not have to be positive or normalized to 1. This brings us to consideration of a wider class of measures.

Definition

The set is called "positive set of measure " if for .

Proposition

If and is a -additive measure then there exists a positive set .

Proof

We are not seeking to obtain a maximal in some sense positive set. The goal is obtain any positive set. Hence, any -style procedure is not necessary.

If does not have a subset of negative measure then we are done. Otherwise we can find a smallest integer such that s.t. (hence, we are trying to grab a set of negative measure with a large absolute value). We set and repeat the procedure. If the procedure does not terminate at finding a positive set then we obtain a disjoint sequence .

We check that is a positive set. By construction, is a non decreasing sequence. Since then . By construction of the set does not have a set with negative measure greater then by absolute value. This means that for any there exists some such that does not have set with negative measure less then . Hence, the set does not have a subset with negative measure.

Proposition

(Hahn decomposition theorem). Let be a signed measure. Then there exist a unique (up to measure-0 sets) pair of a positive set and a negative set such that .

Proof

We seek a maximal positive set and its complement. Either there exists a set s.t. or it exists not. In the later case . In the former case there exists at least one positive set (see the previous proposition). Hence, the is well defined. We select a sequence of positive sets such that . Then we take . is a positive set. The set is a negative set. Indeed, existence of a set of positive measure within leads to contradiction with the construction of .

Definition

Measures and are called "mutually singular" iff The measure is said to be "absolutely continuous" with respect to iff

Proposition

(Jordan decomposition theorem). Let be a signed measure. The there exists a unique (up to sets of measure 0) pair of mutually singular positive measures and such that .

Proof

Using the notation of the previous proposition set and .

Definition

The measure is called "total variation" or "absolute value" of .

Proposition

(Radon-Nikodym theorem) Let are signed measures ( -additive). Then is absolutely continuous with respect to if and only if for some positive measurable function . The function is unique.

Proof

The proposition contains several statements but the construction of based on and is the main idea and the only non-trivial part. For simplicity we assume that is a positive measure (because otherwise we use the proposition ( Hahn decomposition theorem ) to modify the proof). We reconstruct the from values of and according to the following procedure. Observe that the measure , should be of the form . We also have Hahn decomposition ( Hahn decomposition theorem ) at our disposal. Hence, we vary among the numbers at consider the measures . This gives us level sets of with increasing precision as . Then we construct an -indexed sequence of simple functions (see definition ( Simple function definition )). Let are the Hahn decomposition sets of . We set on . Such sequence is almost surely a Cauchy sequence. Hence, it converges to some measurable integrable function . By the dominated convergence ( Dominated convergence theorem ) . Then we estimate for any using the construction.

Notation

The function of the proposition ( Radon-Nikodym theorem ) is denoted

 Notation. Index. Contents.