26
L. H. F. de Andrade et al.
(t, 2u) ≤ K (t, u),
for all u ≥ f (t) and μ-a.e. t ∈ T.
One can see that, if ∈ 2 , then
T (t, |u(t)|) dμ < ∞, for every u ∈ L
, in
other words, it holds for the set of functions u which belong to the Musielak–Orlicz
space. Then, L
, ˜
L
and E
are equal as sets. Otherwise, if the Musielak–Orlicz
function c (u) = ϕ(c(t) + u) − ϕ(c(t)) does not satisfy the 2 -condition, then E
ϕ
c
is a proper subspace of L
ϕ
c . It is easy to see this fact for Orlicz space [6, Theorem 10.1].
Moreover, we have the following result:
Lemma 2.1 [28, Remark 3.12] Let be a Musielak–Orlicz function not satisfying
the 2 -condition and such that (t, b (t)) = ∞ for μ-a.e. (almost everywhere)
t ∈ T , where b (t) = sup{u ≥ 0 : (t, u) < ∞}. Then we can find functions u ∗
and u
∗ in L
such that
I (λu ∗ ) < ∞, for 0 ≤ λ ≤ 1,
I (λu ∗ ) = ∞, for 1 < λ,
(2.7)
and
I (λu
∗
) < ∞, for 0 ≤ λ < 1,
I (λu
∗
) = ∞, for 1 ≤ λ.
,
(2.8)
where I c (u(t)) =
T c (t, |u(t)|)dμ for any u ∈ L
0 .
Another important fact is that, given a deformed exponential ϕ(·) that satisfies the
Condition 2.1, we can always find a Musielak–Orlicz function c (t, u) = ϕ(t, c(t) +
u) − ϕ(t, c(t)) that does not satisfy the 2 -condition. This is stated in the following
Proposition.
Proposition 2.1 [26, Proposition 2] Given any deformed exponential ϕ, we can find
a measurable function c : T → R with
T ϕ(c)dμ = 1 such that the Musielak–Orlicz
function c (t, u) = ϕ(t, c(t) + u) − ϕ(t, c(t)) does not satisfy the 2 -condition.
It is discussed in [26] that, given two Musielak–Orlicz functions c (t, u) =
ϕ(c(t) + u) − ϕ(c(t)) and b (t, u) = ϕ(b(t) + u) − ϕ(b(t)), with b, c : T → R
functions such that
T ϕ(b) dμ = 1 and
T ϕ(c) dμ = 1, that satisfy the
2 -condition, then L
ϕ
c and L
ϕ
b are equal as sets and F
ϕ
c = F
ϕ
b [26, Proposition 4].
In the next section, we will investigate the behavior of the normalizing function
ψ near the boundary of the domain of the parametrization. This is important to keep
the normalization function still valid for a given probability distribution.
L. H. F. de Andrade et al.
(t, 2u) ≤ K (t, u),
for all u ≥ f (t) and μ-a.e. t ∈ T.
One can see that, if ∈ 2 , then
T (t, |u(t)|) dμ < ∞, for every u ∈ L
, in
other words, it holds for the set of functions u which belong to the Musielak–Orlicz
space. Then, L
, ˜
L
and E
are equal as sets. Otherwise, if the Musielak–Orlicz
function c (u) = ϕ(c(t) + u) − ϕ(c(t)) does not satisfy the 2 -condition, then E
ϕ
c
is a proper subspace of L
ϕ
c . It is easy to see this fact for Orlicz space [6, Theorem 10.1].
Moreover, we have the following result:
Lemma 2.1 [28, Remark 3.12] Let be a Musielak–Orlicz function not satisfying
the 2 -condition and such that (t, b (t)) = ∞ for μ-a.e. (almost everywhere)
t ∈ T , where b (t) = sup{u ≥ 0 : (t, u) < ∞}. Then we can find functions u ∗
and u
∗ in L
such that
I (λu ∗ ) < ∞, for 0 ≤ λ ≤ 1,
I (λu ∗ ) = ∞, for 1 < λ,
(2.7)
and
I (λu
∗
) < ∞, for 0 ≤ λ < 1,
I (λu
∗
) = ∞, for 1 ≤ λ.
,
(2.8)
where I c (u(t)) =
T c (t, |u(t)|)dμ for any u ∈ L
0 .
Another important fact is that, given a deformed exponential ϕ(·) that satisfies the
Condition 2.1, we can always find a Musielak–Orlicz function c (t, u) = ϕ(t, c(t) +
u) − ϕ(t, c(t)) that does not satisfy the 2 -condition. This is stated in the following
Proposition.
Proposition 2.1 [26, Proposition 2] Given any deformed exponential ϕ, we can find
a measurable function c : T → R with
T ϕ(c)dμ = 1 such that the Musielak–Orlicz
function c (t, u) = ϕ(t, c(t) + u) − ϕ(t, c(t)) does not satisfy the 2 -condition.
It is discussed in [26] that, given two Musielak–Orlicz functions c (t, u) =
ϕ(c(t) + u) − ϕ(c(t)) and b (t, u) = ϕ(b(t) + u) − ϕ(b(t)), with b, c : T → R
functions such that
T ϕ(b) dμ = 1 and
T ϕ(c) dμ = 1, that satisfy the
2 -condition, then L
ϕ
c and L
ϕ
b are equal as sets and F
ϕ
c = F
ϕ
b [26, Proposition 4].
In the next section, we will investigate the behavior of the normalizing function
ψ near the boundary of the domain of the parametrization. This is important to keep
the normalization function still valid for a given probability distribution.