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1 1) (X) ≥ 0.

2 2) (X) = [X2] − ([X])2.

3 3) For any (a, b) ∈ ℝ2, (aX + b) = a2(X).

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 ssss1.

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.

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

 – 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.