2 On Normalization Functions and ϕ-Families of Probability Distributions
31
T
u n dμ ≥ 2
n
α n ,
for all n ≥ 1,
then an increasing sequence {n i } of natural numbers and a sequence {A i } of pairwise
disjoint, measurable sets can be found, such that
A i
u n i dμ = α n i ,
for all i ≥ 1.
For the next lemma we denote the functional I c =
T c (t, | u(t) |)dμ for any
u ∈ L
0 .
Lemma 2.4 Consider c : T → [0, ∞) a measurable function such that
T ϕ(c)dμ <
∞. Suppose that, for each λ > 0, we cannot find α > 0 and f ∈ ˜
L
c such that
αα c (t, u) ≤ c−λu 0 (t, u),
for all u > f (t).
(2.10)
Then, a strictly decreasing sequence 0 < λ n ↓ 0, and sequences {u n } and {A n } of
finite-value, measurable functions, and pairwise disjoint, measurable sets, respectively, can be found such that
I c (u n χ A n ) = 1, and I c−λn u 0 (u n χ A n ) ≤ 2
−n
,
for all n ≥ 1.
(2.11)
Proof Let {λ m } be a strictly decreasing sequence such that 0 < λ m ↓ 0. Define the
non-negative functions
f m (t) = sup{u > 0 : 2
−m
c (t, u) > > c−λ m u 0 (t, u)}, for all m ≥ 1,
where we adopt the convention that sup ∅ = 0. Since (2.10) is not satisfied, we
have that I c ( f m ) = ∞ for each m ≥ 1. For every rational number r > 0, define the
measurable sets
A m,r = {t ∈ T : 2
−m
c (t, r ) > > c−λ m u 0 (t, r )},
and the simple functions u m,r = r χ A m,r .
For r = 0, we obtain the set u m,r = 0. Let {r i } be an enumeration of the
non-negative rational numbers with r 1 = 0. Define the non-negative, simple functions v m,k = max 1≤i≤k u m,r i , for each m, k ≥ 1. By the continuity of c (t, ·) and
c−λ m u 0 (t, ·), it follows that v m,k ↑ f m as k → ∞. From the Monotone Convergence
Theorem for each m ≥ 1, we can find some k m ≥ 1 such that the function v m = v m,k m
satisfies I c (v m ) ≥ 2
m .
31
T
u n dμ ≥ 2
n
α n ,
for all n ≥ 1,
then an increasing sequence {n i } of natural numbers and a sequence {A i } of pairwise
disjoint, measurable sets can be found, such that
A i
u n i dμ = α n i ,
for all i ≥ 1.
For the next lemma we denote the functional I c =
T c (t, | u(t) |)dμ for any
u ∈ L
0 .
Lemma 2.4 Consider c : T → [0, ∞) a measurable function such that
T ϕ(c)dμ <
∞. Suppose that, for each λ > 0, we cannot find α > 0 and f ∈ ˜
L
c such that
αα c (t, u) ≤ c−λu 0 (t, u),
for all u > f (t).
(2.10)
Then, a strictly decreasing sequence 0 < λ n ↓ 0, and sequences {u n } and {A n } of
finite-value, measurable functions, and pairwise disjoint, measurable sets, respectively, can be found such that
I c (u n χ A n ) = 1, and I c−λn u 0 (u n χ A n ) ≤ 2
−n
,
for all n ≥ 1.
(2.11)
Proof Let {λ m } be a strictly decreasing sequence such that 0 < λ m ↓ 0. Define the
non-negative functions
f m (t) = sup{u > 0 : 2
−m
c (t, u) > > c−λ m u 0 (t, u)}, for all m ≥ 1,
where we adopt the convention that sup ∅ = 0. Since (2.10) is not satisfied, we
have that I c ( f m ) = ∞ for each m ≥ 1. For every rational number r > 0, define the
measurable sets
A m,r = {t ∈ T : 2
−m
c (t, r ) > > c−λ m u 0 (t, r )},
and the simple functions u m,r = r χ A m,r .
For r = 0, we obtain the set u m,r = 0. Let {r i } be an enumeration of the
non-negative rational numbers with r 1 = 0. Define the non-negative, simple functions v m,k = max 1≤i≤k u m,r i , for each m, k ≥ 1. By the continuity of c (t, ·) and
c−λ m u 0 (t, ·), it follows that v m,k ↑ f m as k → ∞. From the Monotone Convergence
Theorem for each m ≥ 1, we can find some k m ≥ 1 such that the function v m = v m,k m
satisfies I c (v m ) ≥ 2
m .