= {∅, Ω, A, Ac} 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,
We will now recall the concept of the product σ-algebra.
DEFINITION 1.4.– Let (Ei,
– Let n ∈ ℕ. The σ-algebra defined over and generated by
is denoted by
In the specific case where E0 = ... = En = E and
– 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
– 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,
In practice, when E ⊂ ℝ, we set ε =
EXAMPLE 1.5.– If (Ω,
is
Thus, in all cases, we do have
EXAMPLE 1.6.– The composition of two measurable functions is measurable. Indeed, if (Ω,
Thus, the composition g ∘ f is indeed measurable on (Ω,
1.2. Probability elements
We will now review the concept of a probability measure or probability distribution, and the concept of random variable, as well as the chief properties of these concepts.
