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We now define the σ-algebra generated by a random variable. This concept is important for several reasons. For instance, it can make it possible to define the independence of random variables. It is also at the heart of the definition of conditional expectations; see Chapter 2.

PROPOSITION 1.6.– Let X be a real random variable, defined on (Ω, , ℙ) taking values in (E, ε ). Then, _X = X⁻¹(ε) = {X⁻¹(A); A ∈ ε} is a sub-σ-algebra of on Ω. This is called the σ-algebra generated by the random variable X. It is written as σ(X). It is the smallest σ-algebra on Ω that makes X measurable:

EXAMPLE 1.19.– Let ₀ = {∅, Ω} and X = c ∈ ℝ be a constant. Then, for any Borel set B in ℝ, (X ∈ B) has the value ∅ if c ∉ B and Ω if c ∈ B. Thus, the σ-algebra generated by X is ₀. Reciprocally, it can be demonstrated that the only ₀-measurable random variables are the constants. Indeed, let X be a ₀-measurable random variable. Assume that it takes at least two different values, x and y. It may be assumed that y ≥ x without loss of generality. Therefore, We have that (X ∈ B) is non-empty because x ∈ B but is not Ω since y ∉ B. Therefore, X is not ₀-measurable.

PROPOSITION 1.7.– Let X be a random variable on (Ω, , ℙ) taking values in (E, ε) and let σ(X) be the σ-algebra generated by X. Thus, a random variable Y is σ(X)-measurable if and only if there exists a measurable function f such that Y = f (X).

This technical result will be useful in certain demonstrations further on in the text. In general, if it is known that Y is σ(X)-measurable, we cannot (and do not need to) make explicit the function f. Reciprocally, if Y can be written as a measurable function of X, it automatically follows that Y is σ(X)-measurable.

EXAMPLE 1.20.– A die is rolled 2 times. This experiment is modeled by Ω = {1, 2, 3, 4, 5, 6}² endowed with the σ-algebra of its subsets and the uniform distribution. Consider the mappings X₁, X₂ and Y from Ω onto ℝ defined by

thus, X_i is the result of the ith roll and Y is the parity indicator of the first roll. Therefore, thus, Y is σ(X₁)-measurable. On the other hand, Y cannot be written as a function of X₂.

The σ-algebra generated by X represents all the events that can be observed by drawing X. It represents the information revealed by X.

DEFINITION 1.14.– Let (Ω, , ℙ) be a probability space.

 – Let X and Y be two random variables on (Ω, , ℙ) taking values in (E1, ε1) and (E2, ε2). Then, X and Y are said to be independent if the σ-algebras σ(X) and σ(Y) are independent.

 – Any family (Xi)i∈I of random variables is independent if the σ-algebras σ(Xi) are independent.

 – Let be a sub-σ-algebra of , and let X be a random variable. Then, X is said to be independent of if σ(X) is independent of or, in other words, and are independent.

PROPOSITION 1.8.– If X and Y are two integrable and independent random variables, then their product XY is integrable and [XY] = [X][Y].