Benoîte de Saporta - Martingales and Financial Mathematics in Discrete Time

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, ), ( E , ε ) and ( G , ) are three probabilizable spaces, f : Ω ↦ E and g : E ↦ G are two ( , ε ) and ( ε , ) -measurable mappings, respectively, then for any B ∈ , g −1( B ) ∈ ε and consequently ,

Thus, the composition g ∘ f is indeed measurable on (Ω, ) in ( G , ).

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.

A probability measure or probability distribution is a finite measure whose total mass is equal to 1.

DEFINITION 1.7.– A probability or probability measure, or law of probability or distribution over a probability space (Ω, ) is a measure with a total mass equal to 1. In other words, a probability over (Ω, ) is a mapping ℙ : → ℝ such that

– for any A ∈ , ℙ(A) ≥ 0,

– ℝ(Ω) = 1,

– for any sequence of pairwise disjoint events in , denoted by (An)n∈ℕ, we have

The triplet (Ω, , ℙ) is then called a probability space .

EXAMPLE 1.7.– Ω is endowed with the coarse σ-algebra = {∅, Ω} . Thus, the single probability measure on (Ω, ) is given by:

EXAMPLE 1.8.– Let Ω = [0, 1] and = = ([0, 1]) be the Borel σ-algebra of [0, 1] . If λ denotes the Lebesgue measure, then the mapping:

is a probability measure on (Ω, ).

EXAMPLE 1.9.– Let Ω be non-empty set such that card (Ω) < ∞, where card (Ω) denotes the cardinal of Ω, that is, the number of elements in Ω . Consider the mapping ℙ from (Ω) onto [0, 1] such that for every

The mapping ℙ is then a probability on (Ω, (Ω)), said to be the uniform probability on Ω.

We will only review those properties of a probability that will be useful for this book.

PROPOSITION 1.2.– Let (Ω, , ℙ) be a probability space and ( A n) n∈ℕ be a sequence of events in .

– If (An)n∈ℕ is increasing (for the inclusion), then,