2 On Normalization Functions and ϕ-Families of Probability Distributions
25
E
ϕ
c =
⎧
⎨
⎩
u ∈ L
0
;
T
ϕ(c(t) + λu(t)) dμ < ∞ for all λ > 0
⎫
⎬
⎭
.
Clearly, E
ϕ
c ⊆ ˜
L
ϕ
c ⊆ L
ϕ
c .
Now, let K
ϕ
c be the convex set of all the functions u ∈ L
ϕ
c such that ϕ(c + λu) is
μ-integrable for every λ in a neighborhood of [0, 1], that is,
K
ϕ
c =
⎧
⎨
⎩
u ∈ L
ϕ
c ;
T
ϕ(c + λu) < ∞, for some λ > 1
⎫
⎬
⎭
.
We know that K
ϕ
c is an open set in L
ϕ
c [27, Lemma 2] and, for u ∈ K
ϕ
c , the function
ϕ(c + u) is not necessarily in P μ . Hence, the normalizing function ψ : K
ϕ
c → R is
introduced in order to make the density ϕ(c + u − ψ(u)u 0 ) to be in P μ [27]. For any
u ∈ K
ϕ
c , ψ(u) ∈ R is the unique function which ϕ(c + u − ψ(u)u 0 ) is in P μ [27,
Proposition 3].
Let
B
ϕ
c =
⎧
⎨
⎩
u ∈ L
ϕ
c :
T
uϕ
+ (t, c(t))dμ = 0
⎫
⎬
⎭
be a closed subspace of L
ϕ
c , thus for every u ∈ B
ϕ
c = B
ϕ
c ∩ K
ϕ
c , by the convexity of
ϕ, one has ψ(u) ≥ 0 and ϕ(c + u − ψ(u)u 0 ) ∈ P μ .
For each measurable function c : T → R such that p = ϕ(c) ∈ P μ is associated
a parametrization ϕ c : B
ϕ
c → F
ϕ
c , given by
ϕ c (u) = ϕ(c + u − ψ(u)u 0 ),
(2.6)
where the operator ϕ acts on the set of real-value functions u : T → R given by
ϕ(u)(t) = ϕ(u(t)) and the set F
ϕ
c = ϕ c (B
ϕ
c ) ⊆ P μ where P μ =
{F
ϕ
c : ϕ(c) ∈ P μ }
and the map ϕ c is a bijection from B
ϕ
c to F
ϕ
c .
In the following section, we will discuss the 2 -condition, which is a condition
that the Musielak–Orlicz functions can satisfy or not. We will discuss this condition
and its consequences.
2.2.3 The 2 - Condition and ϕ-Families of Probability
Distributions
Condition 2.3 ( 2 condition) A Musielak–Orlicz function satisfies the 2 -
condition ( ∈ 2 ), if one can find a constant K > 0 and a non-negative function
f ∈ ˜
L
ϕ
c such that