Lecture Notes - Department of Mathematics and Statistics - Queen's ...

4 CHAPTER 1. REVIEW OF PROBABILITY
simple functions h n such that h n (x) → h(x) monotonically, that is h n+1 (x) ≥ h n (x). We define the limit as the
Lebesgue integral:
∫
n∑
lim h n (dx)µ(dx) = lim b k µ(A k )
n→∞
n→∞
We note that the space of simple functions is dense in the space of bounded measurable functions on R under
the supremum norm.
There are three important convergence theorems which we will not discuss in detail, the statements of which will
be given in class.
k=1
1.2.6 Fatou’s Lemma, the Monotone Convergence Theorem and the Dominated
Convergence Theorem
Theorem 1.2.3 (Monotone Convergence Theorem) If µ is a σ−finite positive measure on (X, B(X)) and
{f n , n ∈ Z + } is a sequence of measurable functions from X to R which pointwise, monotonically, converge to f:
that is, 0 ≤ f n (x) ≤ f n+1 (x) for all n, and
lim f n(x) = f(x),
n→∞
for µ−almost every x, then
∫
X
∫
f(x)µ(dx) = lim f n (x)µ(dx)
n→∞
X
Theorem 1.2.4 (Fatou’s Lemma) If µ is a σ−finite positive measure on (X, B(X)) and {f n , n ∈ Z + } is a
sequence of non-negative measurable functions from X to R, then
∫
∫
liminf f n(x)µ(dx) ≤ liminf f n (x)µ(dx)
n→∞ n→∞
X
Theorem 1.2.5 (Dominated Convergence Theorem) If (i) µ is a σ−finite positive measure on (X, B(X)),
(ii) g(x) ≥ 0 is a Borel measurable function such that
∫
g(x)µ(dx) < ∞,
X
and (iii) {f n , n ∈ Z + } is a sequence of measurable functions from X to R which satisfy |f n (x)| ≤ g(x) for
µ−almost every x, and lim n→∞ f n (x) = f(x), then:
∫
∫
f(x)µ(dx) = lim f n (x)µ(dx)
n→∞
X
Note that for the monotone convergence theorem, there is no restriction on boundedness; whereas for the dominated
convergence theorem, there is a boundedness condition. On the other hand, for the dominated convergence
theorem, the pointwise convergence does not have to be monotone.
X
X
1.3 Probability Space and Random Variables
Let (Ω, F) be a measurable space. If P is a probability measure, then the triple (Ω, F, P) forms a probability
space. Here Ω is a set called the sample space. F is called the event space, and this is a σ−field of subsets of Ω.
Let (E, E) be another measurable space and X : (Ω, F, P) → (E, E) be a measurable map. We call X an
E−valued random variable. The image under X is a probability measure on (E, E), called the law of X.
The σ-field generated by the events {{w : X(w) ∈ A}, A ∈ E} is called the σ−field generated by X and is denoted
by σ(X).