Theory of Statistics - George Mason University

726 0 Statistical Mathematics
Proof.
Assume the hypothesis: that is, fn ↗ f for all ω ∈ A where ν(Ac ) =
0. Now restrict each function to A, and observe that fnIA ↗ fIA and
fnIA dν = fn dν and fIA dν = f dν. Lemma 0.1.23.1 immediately
implies fn dν ↗ f dν.
That Theorem 0.1.23 follows so readily from Lemma 0.1.23.1 is another
illustration of a principle of real analysis stated by Littlewood that every convergent
sequence of functions is “nearly” uniformly convergent (see page 753).
In the hypotheses of the lemma, we have only pointwise convergence. Without
needing uniform convergence, however, we extend the conclusion to the case
of convergence a.e.
Theorem 0.1.24 (Fatou’s lemma)
For nonnegative integrable Borel fn,
liminf fn dν ≤ liminf n n
fn dν. (0.1.49)
Proof.
Let gn = infk≥n fk and g = limn inf fn. As in the monotone convergence
theorem, gn is nonnegative and gn ↗ g, so gn dν ↗ g dν. Also, for each
n, fn dν ≥ gn dν; hence, we have the desired conclusion.
The next theorem is the most powerful of the convergence theorems for
integrals.
Theorem 0.1.25 (Lebesgue’s dominated convergence)
If limn→∞ fn = f a.e. and there exists an integrable function g such that
|fn| ≤ g a.e., then
Proof.
***
lim
n→∞
fn dν = f dν. (0.1.50)
Corollary 0.1.25.1 (bounded convergence)
Let {fn} be a sequence of measurable functions defined on a set A, where
ν(A) < ∞. If for some real number M, |fn(ω)| ≤ M, and limn→∞ fn(ω) =
f(ω) for each ω ∈ A then
lim fn dν = f dν. (0.1.51)
n→∞
A A
0.1.6.5 Integrals over More than One Measure Space
So far the integrals we have discussed have been for functions over a single
measure space. We now consider integrals over more than one measure space.
Theory of Statistics c○2000–2013 James E. Gentle