2 On Normalization Functions and ϕ-Families of Probability Distributions
33
T
ϕ(c + u − λu 0 )dμ =
B
ϕ(c − λu 0 )dμ +
∞
n=n 0
A n
ϕ((c − λu 0 ) + u n )dμ
=
B
ϕ(c − λu 0 )dμ
+
∞
n=n 0
⎧
⎨
⎩
A n
ϕ(c − λu 0 )dμ + I c−λu 0 (u n χ A n )
⎫
⎬
⎭
≤
T
ϕ(c − λu 0 )dμ +
∞
n=n 0
2
−n
< ∞.
Consequently, for α ∈ (0, 1), we can write
T
ϕ(c + αu)dμ =
T
ϕ
c + α(u − λu 0 ) + (1 − α)
αλ
1 − α
u 0
dμ
≤ α
T
ϕ(c + u − λu 0 )dμ + (1 − α)
T
ϕ
c +
αλ
1 − α
u 0
dμ
< ∞.
On the other hand, for α ≥ 1, it follows that
T
ϕ(c + αu)dμ ≥
B
ϕ(c)dμ +
∞
n=n 0
A n
ϕ(c + u n )dμ
≥
B
ϕ(c)dμ +
∞
n=n 0
⎧
⎨
⎩
A n
ϕ(c)dμ + I c (u n χ A n )
⎫
⎬
⎭
=
T
ϕ(c)dμ +
∞
n=n 0
1 = ∞.
We can choose λ
< 0 such that
w = λ
u 0 χ B +
∞
n=n 0
u n χ A n
satisfies
T wϕ
+ (c)dμ = 0. Clearly,
T ϕ(c + w)dμ = ∞,
T ϕ(c + αw)dμ < ∞
for α ∈ (0, 1) and
T ϕ(c + αw)dμ = ∞ for α > 1, that is, w ∈ ∂B
ϕ
c and
T ϕ(c +
w − λu 0 )dμ < ∞ for some fixed λ > 0.