Читать книгу Martingales and Financial Mathematics in Discrete Time онлайн

12 страница из 24

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.

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

i

ii∈ℕ

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

n

n

n

 – 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.

 – 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) < ∞.