Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

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 =

-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,

),