Читать книгу Martingales and Financial Mathematics in Discrete Time - Benoîte de Saporta - Страница 9

Let us start by reviewing the concept of a σ-algebra.

DEFINITION 1.1.– A subset of (Ω) is a σ-algebra over Ω if

1 1) Ω ∈ ;

2 2) is stable by complementarity: for any A ∈ , we have Ac ∈ , where Ac denotes the complement of A in Ω: Ac = Ω\A;

3 3) is stable under a countable union: for any sequence of elements (An)n∈ℕ of , we have

Elements of a σ-algebra are called events.

EXAMPLE 1.1.– The set = {∅, Ω} is a σ-algebra and is also the smallest σ-algebra over Ω; it is called the trivial σ-algebra. Indeed, is in fact a σ-algebra since Ω ∈ and by creating unions of ∅ and Ω we always obtain ∅ ∈ or Ω ∈ . Further, for any other σ-algebra , we clearly have ⊂ .

EXAMPLE 1.2.– The set (Ω) is the largest σ-algebra over Ω; it is called the largest σ-algebra. Indeed, by construction, (Ω) contains all the subsets of Ω, and thus it contains in particular Ω and it is stable by complementarity and under countable unions. In addition, any other σ-algebra over Ω is clearly included in (Ω).

DEFINITION 1.2.– Let Ω be a non-empty set and be a σ-algebra over Ω. The couple (Ω, ) is called a probabilizable space.

Among the elementary properties of σ-algebra, we can cite stability through any intersection (countable or not).

PROPOSITION 1.1.– Any intersection of σ-algebras over a set Ω is a σ-algebra over Ω.

PROOF.– Let (_i)_i∈I be any family of σ-algebra indexed by a non-empty set I. Thus,

 – first of all, for any i, Ω ∈ i, thus Ω ∈ ∩i∈Ii;

 – secondly, if A ∈ ∩i∈I i, then for any i, A ∈ i. As these are σ-algebras, we have that for any i, Ac ∈ i, thus Ac ∈ ∩i∈I i;

 – finally, if for any n ∈ ℕ, An ∈ ∩i∈I i, then for any i, n, An ∈ i. As these are σ-algebras, we have that for any i, ∪n∈ℕAn ∈ i, thus

It is generally difficult to make explicit all the events in a σ-algebra. We often describe it using generating events.

DEFINITION 1.3.– Let ε be a subset of (Ω). The σ-algebra σ(ε) generated by ε is the intersection of all σ-algebras containing ε. It is the smallest σ-algebra containing ε. ε is called the generating system of the σ-algebra σ(ε).

It can be seen that σ(ε) is indeed a σ-algebra, being an intersection of σ-algebras.

EXAMPLE 1.3.– If A ⊂ Ω, then, σ(A) = {∅, Ω, A, A^c} is the smallest σ-algebra Ω containing A.

EXAMPLE 1.4.– If Ω is a topological space, the σ-algebra generated by the open sets of Ω is called the Borel σ-algebra of Ω. A Borel set is a set belonging to the Borel σ-algebra. On ℝ, (ℝ) generally denotes the σ-algebra of Borel sets. It must be recalled that this is also the σ-algebra generated by the intervals, or by the intervals of the form ] − ∞, x], x ∈ ℝ. Thus, there is no unicity of the generating system.

We will now recall the concept of the product σ-algebra.

DEFINITION 1.4.– Let (E_i, _i)_i∈ℕ be a sequence of measurable spaces.

 – Let n ∈ ℕ. The σ-algebra defined over and generated by

is denoted by ₀ ⊗ ... ⊗ _n, and it is called the product σ-algebra over We have, in particular,

In the specific case where E₀ = ... = E_n = E and ₀ = ... = _n = , we also write

 – We use ⊗i∈ℕi to denote the σ-algebra over the countable product space generated by the sets of the form where Ai ∈ i and Ai = Ei except for a finite number of indices i. In the specific case where, for any and i = , the product space is denoted by Eℕ, and the σ-algebra ⊗i∈ℕi is denoted by ⊗N.

Finally, let us review the concepts of measurability and measure.

DEFINITION 1.5.– Let Ω be non-empty set and be a σ-algebra on Ω.

 – A measure over a probabilizable space (Ω, ) is defined as any mapping μ defined over , with values in [0, +∞] = ℝ+ ∪ {+∞}, such that μ(∅) = 0 and for any family (Ai)i∈ℕ of pairwise disjoint elements of , we have the property of σ-additivity:

 – A measure μ over a probabilizable space (Ω, ) is said to be finite, or have finite total mass, if μ(Ω) < ∞.

 – If μ is a measure over a probabilizable space (Ω, ), then the triplet (Ω, , μ) is called a measured space.

DEFINITION 1.6.– Let (Ω, ) and (E, ε ) be two probabilizable spaces. A mapping X, defined over Ω taking values in E, is said to be (, ε)-measurable, or just measurable, if there is no ambiguity regarding the reference σ-algebras, if

In practice, when E ⊂ ℝ, we set ε = (E) the set of Borel subsets of E, that is, the set of subsets of E. We can simply say that X is -measurable. When, in addition, we manipulate a single σ-algebra over Ω, it can be simply said that X is measurable. If we work with several σ-algebras over Ω, the concerned σ-algebra must always be specified: X is -measurable.

EXAMPLE 1.5.– If (Ω, ) is a measurable space and A ∈ , then the indicator function

is -measurable. Indeed, for any Borel set B in ℝ, we have

Thus, in all cases, we do have

EXAMPLE 1.6.– The composition of two measurable functions is measurable. Indeed, if (Ω, ), (E, ε) and (G, ) are three probabilizable spaces, f : Ω ↦ E and g : E ↦ G are two (, ε) and (ε, )-measurable mappings, respectively, then for any B ∈ , g⁻¹(B) ∈ ε and consequently,

Thus, the composition g ∘ f is indeed measurable on (Ω, ) in (G, ).