Abstract
Very well known and important is the theory of the existence and uniqueness of measures invariant under a shift on a group (so-called Haar measure) in some groups. It was studied in many spaces and transformations. Such measure m is defined on a family \(\mathcal {F}\) of sets and such that \(m(T^{-1}(A))=m(A)\) for any \(A\in \mathcal {F}.\) In the paper instead of sets intuitionistic fuzzy sets (IF-sets) are studied. As a special case the theory of invariant measures on fuzzy sets can be obtained.
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1 Introduction
In the classical measure theory it is known theory about Haar measure [4] stating that in every compact Abelian group there exists a probability measure invariant under shifts.
Let \((G,+)\) be a compact Abelian topological group, \(\mathcal C\) be the family of all compact subsets of \(G, \sigma (\mathcal C)\) be the \(\sigma \)-algebra generated by \(\mathcal C\). Then there exists exactly one probability measure \(P:\sigma (\mathcal C) \rightarrow [0,1]\) such that
for any \(A \in \sigma (\mathcal C)\) and any \(a \in G\) (see e.g. [4]). The measure P is usually called the Haar measure or invariant probability measure. Recall that in [8] a version of the existence of invariant measure has been proved for semigroups, and in [9] for IP-loops.
It is natural to consider fuzzy sets instead of sets [10]. In the paper we shall study the theory of invariant measures on families of intuitionistic fuzzy sets [1]. Recall that in the paper [5] there was considered a special case, the group \((R,+)\), and the shitf \(T_a:[0,1) \rightarrow [0,1)\) given by the prescription \(T_a(x) = x + a (mod 1)\). Then the theorem about the existence of an invariant IF-states on real numbers was proved. In this paper we will make an extension of this theory to compact Abelian topological group and we will use more general invariant transformation.
In the paper we shall prove the existence of an invariant measures on the family of intuitionistic fuzzy sets [1].
An intutionistic fuzzy set (IFS) is a pair \(A = (\mu _A, \nu _A)\) of functions \(\mu _A, \nu _A : G \rightarrow [0,1]\) such that
If \(A = (\mu _A, \nu _A), B = (\mu _B, \nu _B)\), then we write
if and only if
Here \((0_G,1_G) \le (\mu _A, \nu _A) \le \ (1_G,0_G)\) for all \(A = (\mu _A, \nu _A)\). We shall write
if and only if
Denote by \(\bigtriangleup \) the set
Then an IF-set is a mapping \(A:G \rightarrow \bigtriangleup \). If we put \(\nu _A = 1 - \mu _A\), then we obtain a fuzzy set \(A:G \rightarrow [0,1]\). If \(A:G \rightarrow \{0,1\}\), then we obtain a crisp subset \(A_0 \subset G\), where \(\omega \in A_0\) if and only if \(A(\omega ) = 1\), hence A can be identified with the indicator \(\chi _{A_0}\).
In the paper we shall work with the family \(\mathcal F\) of all \(A = (\mu _A, \nu _A):G \rightarrow \bigtriangleup \) with \(\mu _A, \nu _A\) continuous. The Lukasiewicz binary operations are defined on \(\mathcal F\) by the following way
By a state on \(\mathcal F\) we consider a mapping \(m:\mathcal F \rightarrow [0,1]\) satisfying the following conditions:
-
1.
\(m((0_G,1_G)) = 0, m((1_G, 0_G)) = 1\);
-
2.
\(A \odot B = (0_G,1_G) \Longrightarrow m(A \oplus B) = m(A) + m(B)\);
-
3.
\(A_n \nearrow A \Longrightarrow m(A_n) \nearrow m(A)\).
Theorem 1
To any state \(m:\mathcal F \rightarrow [0,1]\) there exists a probability measure P defined on the \(\sigma \)-algebra \(\sigma (\mathcal C)\) and there exists \(\alpha \in [0,1]\) such that
for any \(A = (\mu _A,\nu _A) \in \mathcal F\).
Proof
2 Invariant States
Consider \(a \in G\) and define the transformation
by the formula
For this transformation we will use the notation
Example 1
Let us define the transformation \(T_a:[0,1) \rightarrow [0,1)\) by the formula
i.e.
Then the function
is an example of the mentioned transformation on \(\mathcal {F}\). The function \(T_a\) represents the moving around the circle with the circuit equal to one.
Our main result is contained in the following theorem.
Theorem 2
To any \(\beta \in [0,1]\) there exists exactly one state \(m: \mathcal F \rightarrow [0,1]\) such that
for any \(A \in \mathcal F\) and any \(a \in G\) and such that
Proof
Let \(A = (\mu _A, \nu _A) \in \mathcal F\). Let \(P:\sigma (\mathcal C) \rightarrow [0,1]\) be the invariant probability measure, i.e. \(P(B + a) = P(B)\) for any \(B \in \sigma (\mathcal C)\) and any \(a \in G\). Put
Then
for any \(A \in \mathcal F\). We have proved the existence of an invariant state \(m:\mathcal F \rightarrow [0,1]\). Evidently \(m((0_G, 0_G)) = \beta \).
We shall prove the uniqueness. Let \(\lambda : \mathcal F \rightarrow [0,1]\) be any invariant state such that \(\lambda (0_G,0_G) = \beta \). Then by Theorem 1 there exist \(\alpha \in [0,1]\) and a probability measure \(P:\sigma (\mathcal C)\rightarrow [0,1]\) such that
for any \(A \in \mathcal F\).
Put \(\mu _A = 0_G, \nu _A = 0_G\). Then
hence \(\alpha = \beta \).
First let \(\alpha = 0\). Then
Of course, also
since \(\lambda \) is the invariant probability measure then
for any \(A \in \mathcal F, a \in G\). For any \(B \in \sigma (\mathcal C)\) put \(\mu _A = \chi _B\). It follows
hence \(P:\sigma (\mathcal C)) \rightarrow [0,1]\) is invariant. Moreover,
hence P is an invariant probability measure, and it is determined uniquely.
Let now \(\alpha \in (0,1]\). Then
Evidently
hence
Moreover,
Put \(A = (0_G, \nu _A)\). Then
hence
for any \(A \in \mathcal F\) and any \(a \in G\). It is clear that \(P:\sigma (\mathcal C)) \rightarrow [0,1]\) is an invariant measure. Moreover,
Since \(\alpha > 0\), we have
hence \(P:\sigma (\mathcal C) \rightarrow [0,1]\) is the unique invariant probability measure. \(\square \)
3 Conclusion
We have proved for any real number \(\alpha \in [0,1]\) the existence of a unique state \(m:\mathcal F \rightarrow [0,1]\) invariant with respect to the group transformations
and such that
Of course, for different numbers \(\alpha \) we can obtained different states m.
On the other hand for fuzzy sets [10] we have \(\nu _A = 1 - \mu _A\), hence
and
We have obtained the existence of an invariant fuzzy state m, and even unique, it does not depend on \(\alpha \).
So from IF-invariant theory one can obtain the fuzzy invariant theory [11], but the opposite direction is not possible, the family of IF states is more rich. Hence the result for IF sets is not a corollary of the existence of fuzzy invariant state.
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Michalíková, A., Riečan, B. (2018). On Invariant Measures on Intuitionistic Fuzzy Sets. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 642. Springer, Cham. https://doi.org/10.1007/978-3-319-66824-6_46
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DOI: https://doi.org/10.1007/978-3-319-66824-6_46
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