24
L. H. F. de Andrade et al.
Table 2.1 Some examples of deformed exponentials
Function
Expression
Kaniadaki’s κ-exponential [5]
exp κ (x) =
⎧
⎨
⎩
κ x +
√
1 + κ 2 x 2
1
κ , κ = 0
exp(x),
κ = 0
Tsallis exponential [21]
exp q (x) = [1 + (1 − q)x]
1/(1−q)
+
, q ∈ [0, ∞)
Newton exponential [4]
exp φ (x) =
1+αx
1−αx
1
2α , −
1
α < x <
1
α
2.2.2 ϕ-Families of Probability Distributions
In this section, we recall the construction of the ϕ-families of probability distributions
in [27]. Let (T, Σ, μ) be a σ - finite, non-atomic measure space on which probability
distributions are defined. We denote by P μ the family of all probability on T that are
equivalent to measure μ, that is,
P μ =
⎧
⎨
⎩
p ∈ L
0
; p > 0 and
T
p dμ = 1
⎫
⎬
⎭
,
where L
0 is the linear space of all real-value. In this space, T may be seen of as the set
of real numbers R. These families are based on the replacement of the exponential
function by a deformed exponential ϕ(·) that satisfies the condition provided in
Eq. (2.1).
The ϕ-families of probability distributions were built based on the Musielak–
Orlicz spaces [11]. Let ϕ be a deformed exponential. The Musielak–Orlicz function
is defined in [27] by
c (t, u) = ϕ(c(t) + u) − ϕ(c(t)),
(2.5)
where c : T → R is a measurable function such that ϕ(c) is μ-integrable. Then we
have, the Musielak–Orlicz space L
c , the Musielak–Orlicz class ˜
L
c and the MorseTransue space E
c , denoted, respectively, by L
ϕ
c , ˜
L
ϕ
c and E
ϕ
c , which correspond to
the following sets:
L
ϕ
c =
⎧
⎨
⎩
u ∈ L
0
;
T
ϕ(c(t) + λu(t)) dμ < ∞, for every λ ∈ (− ,)
⎫
⎬
⎭
,
˜
L
ϕ
c =
⎧
⎨
⎩
u ∈ L
0
;
T
ϕ(c(t) + u(t)) dμ < ∞
⎫
⎬
⎭
and