Random Variables

1 Measurable Functions

Let (S, Σ ), (Q, Φ ) be measurable spaces. A function h: S → Q is measurable (w.r.t. Σ , Φ ) ...if h^–1(A) ∊ Σ, ...∀ A ∊ Φ.
[Write h^–1(Φ) ⊆ Σ ].

The map h^–1 preserves set operations:

• h^–1(∪_j A_j) = ∪_j h^–1(A_j)
• h^–1(∩_j A_j) = ∩_j h^–1(A_j)
• h^–1(A^c) = (h^–1(A))^c
• etc.

So if h is measurable, the range of h^–1, that is h^–1(Φ), is itself a sub-σ-algebra of Σ.

2 Real-valued Measurable Functions

We will now consider functions such as h: S → IR, ...which are measurable (w.r.t. Σ, B).
Write mΣ for the class of Σ-measurable functions on S, and (mΣ)⁺ for the non-negative elements of mΣ.

2.1 Borel Functions

h: S → IR, ...S topological space, is called Borel if h is B(S)-measurable. Most important case is h: IR → IR

2.2 Propositions on Measurability

• If ...C ⊆ B, ...and σ(C) = B, then ...h^–1: C → Σ ...⇒ ...h ∊ mΣ
• S topological space and ...h: S → IR ...continuous ...then h is Borel
• For any measurable space (S, Σ) a function ...h: S → IR ...is Σ-measurable if

  { h ≤ c } := { s ∊ S | h(s) ≤ c } ∊ Σ ......(∀ c ∊ IR)

2.3 Sums and Products of Measurable Functions

mΣ is and algebra over IR, i.e. if λ ∊ IR and h₁, h₂, h₃ ∊ mΣ, then:

• h₁ + h₂ ∊ mΣ, ...h₁h₂ ∊ mΣ, ...λ ...h₁ ∊ mΣ

3 Composition Lemma

 

4 Measurability of Limits

 

5 Random Variables

(Ω, F, P) probability space, (Q, Φ ) measurable space. A random variable is a measurable function ...X: (Ω, F ) → (Q, Φ ).
(Q,Φ ) is called the state space or range space of the random variable.

6 Real-valued Random Variables

From now on we will be dealing primarily with the special case (Q, Φ ) = ( IR, B )

7 Distribution of Random Variable

X: (Ω, F ) → (Q,Φ ) random variable. The distribution, or law of X (with respect to P) is the function P^X : Φ → [0,1],

P^X (A) ...= ...P ({ω | X(ω) ∊ A}) ...= ...P (X^–1(A)) This is sometimes written in the form P(X ∊ A).

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