The fuzzy analogies of some ergodic theorems
- Anna Tirpáková^{1} and
- Dagmar Markechová^{1}Email author
https://doi.org/10.1186/s13662-015-0499-2
© Tirpáková and Markechová 2015
Received: 27 March 2015
Accepted: 13 May 2015
Published: 10 June 2015
Abstract
In the paper (Markechová in Fuzzy Sets Syst. 48:351-363, 1992), fuzzy dynamical systems have been defined. In this contribution, using the method of F-σ-ideals, we prove analogies of some ergodic theorems for fuzzy dynamical systems.
Keywords
MSC
1 Introduction
In the classical probability theory, which is based on the Kolmogorov axiomatic system [1], a random event is every element of the σ-algebra S of subsets of a set X. A probability is a normalized measure defined on the σ-algebra S. The notion of a σ-algebra S of random events and the concept of a probability space \((X, S, P)\) are the basis of the classical concept of probability theory. In doing so, the event in the classical probability theory is understood as an exactly defined phenomenon and from a mathematical point of view (as mentioned above) it is a classical set. In real life, however, we often talk about events that carry important information, but they are less exact. For example, ‘tomorrow will be nice’, ‘a large number will be scored’, ‘the patient’s condition improved’ are vaguely defined events, so-called fuzzy events. Their probability can be studied using the apparatus of fuzzy sets theory. The first attempts to develop a concept of fuzzy events and their probability came from the founder of fuzzy sets theory, Zadeh [2, 3]. Assuming that the probability space \((X, S, P)\) is given, Zadeh defined a fuzzy event as any S-measurable function \(f:X \to \langle 0, 1 \rangle\) and the probability \(p(f)\) of a fuzzy event f by the formula \(p(f) = \int_{X} f\, dP\). An axiomatic approach to the creation of a probability concept of fuzzy events was devised by Klement [4]. Generally the Klement probability cannot be represented by the Zadeh construction; necessary and sufficient conditions were given by Klement et al. in [5]. A different approach was found in [6–8]. The object of our studies in [9–11] was the fuzzy probability space \((X,M,m)\) defined by Polish mathematician K Piasecki. In [12], the concept of a fuzzy dynamical system was introduced. By a fuzzy dynamical system we understand a system \((X, M, m, \tau)\), where \((X, M, m)\) is any fuzzy probability space and \(\tau: M \to M\) is an m-invariant σ-homomorphism. Fuzzy dynamical systems include the classical dynamical systems; on the other hand they enable one to study more general situations, for example, Markov’s operators. Note that the other approaches to a fuzzy generalization of notion of dynamical system can be found in [13] and [14]. In the papers [12, 15] (see also [16]), we defined the entropy of fuzzy dynamical systems, and using the method of F-σ-ideals we proved a fuzzy version of Kolmogorov-Sinai theorem on generators. Our aim in this contribution is to prove analogies of the following Mesiar ergodic theorems for the case of a fuzzy dynamical system \((X, M, m, \tau)\).
Theorem 1.1
[17]
Theorem 1.2
[17]
In Section 3 we give fuzzy analogies of the above results. In the proofs we will use the method of F-σ-ideals. Note that the first authors, who were interested in the ergodic theory on fuzzy measurable spaces, were Harman and Riečan [18]. They proved the validity of Birkhoff’s individual ergodic theorem [19] for the compatible case.
2 Fuzzy probability spaces and fuzzy dynamical systems
First, we recall the definitions of basic notions and some facts which will be used in the following.
Definition 2.1
[20]
- (P1)
\(m(a \vee a') = 1\) for every \(a \in M\);
- (P2)
if \(\{ a_{n} \}_{n = 1}^{\infty}\) is a sequence of pairwise W-separated fuzzy subsets from M (i.e., \(a_{i} \le a'_{j}\) for \(i \ne j\)), then \(m( \bigvee_{n = 1}^{\infty} a_{n}) = \sum_{n = 1}^{\infty} m(a_{n})\).
The operations with fuzzy sets are defined here by Zadeh [2], i.e., the union of fuzzy subsets a, b of X is a fuzzy set \(a \vee b\) defined by \((a \vee b)(x) = \sup (a(x), b(x))\) for all \(x \in X\) and the intersection of fuzzy subsets a, b of X is a fuzzy set \(a \wedge b\) defined by \((a \wedge b)(x) = \inf (a(x), b(x))\) for all \(x \in X\). The complement of a fuzzy subset a of X is the fuzzy set \(a'\) defined by \(a'(x) = 1 - a(x)\) for all \(x \in X\). The difference of fuzzy subsets a, b of X is the fuzzy set \(a - b: = a \wedge b'\). The partial ordering relation ≤ is defined in the following way: for every \(a, b \in M\), \(a \le b\) if and only if \(a(x) \le b(x)\) for all \(x \in X\). Using the complementation \(':a \to a'\) for every fuzzy subset \(a \in M\), we see that the complementation ′ satisfies two conditions: (i) \((a')' = a\) for every \(a \in M\); (ii) if \(a \le b\), then \(b' \le a'\). So, M is a distributive σ-lattice with the complementation ′ for which the de Morgan laws hold: \(( \bigvee_{n = 1}^{\infty} a_{n})' = \bigwedge_{n = 1}^{\infty} a'_{n}\) and \(( \bigwedge_{n = 1}^{\infty} a_{n})' = \bigvee_{n = 1}^{\infty} a'_{n}\) for any sequence \(\{ a_{n} \}_{n = 1}^{\infty} \subset M\). A couple \((X,M)\), where X is a nonempty set and M is a fuzzy σ-algebra of fuzzy subsets of X, is called a fuzzy measurable space. The fuzzy set \((1 / 2)_{X}\) is defined by \((1 / 2)_{X} = 1 / 2\) for all \(x \in X\). The empty fuzzy set \(0_{X}\) is defined by \(0_{X}(x) = 0\) for all \(x \in X\). The complement of empty fuzzy set is a fuzzy set \(1_{X}\) defined by the equality \(1_{X}(x) = 1\) for all \(x \in X\). It is called a universum. Fuzzy subsets a, b of X such that \(a \wedge b=0_{X}\) are called separated fuzzy sets. Analogous weak notions (W-notions) were defined by Piasecki in [21] as follows: each fuzzy subset \(a \in M\) such that \(a \ge a'\) is called a W-universum; each fuzzy subset \(a \in M\) such that \(a \le a'\) is called a W-empty set. Fuzzy subsets \(a, b \in M\) such that \(a \le b'\) are called W-separated. It can be proved that a fuzzy set \(a \in M\) is a W-universum if and only if there exists a fuzzy set \(b \in M\) such that \(a = b \vee b'\). Each mapping \(m:M \to \langle 0, \infty )\) having the properties (P1) and (P2) is called in the terminology of Piasecki a fuzzy P-measure. Any fuzzy P-measure has the properties analogous to the properties of classical probability measure.
Definition 2.2
[12]
By a fuzzy dynamical system we shall mean a quadruplet \((X, M, m, \tau)\), where \((X, M, m)\) is a fuzzy probability space and \(\tau: M \to M\) is an m-invariant σ-homomorphism, i.e., \(\tau (a') = (\tau (a))'\), \(\tau ( \bigvee_{n = 1}^{\infty} a_{n}) = \bigvee_{n = 1}^{\infty} \tau (a_{n})\) and \(m(\tau (a)) = m(a)\), for every \(a \in M\) and any sequence \(\{ a_{n} \}_{n = 1}^{\infty} \subset M\).
An analog of a random variable from the classical probability theory is an F-observable.
Definition 2.3
[22]
- (i)
\(x(E^{C}) = 1_{X} - x(E)\) for every \(E \in B(R^{1})\);
- (ii)
\(x( \bigcup_{n = 1}^{\infty} E_{n}) = \bigvee_{n = 1}^{\infty} x(E_{n})\) for any sequence \(\{ E_{n} \}_{n = 1}^{\infty} \subset B(R^{1})\),
It is easy to see that if x is an F-observable, then the range of the F-observable x, i.e., the set \(R(x): = \{ x(E); E \in B(R^{1})\}\), is a Boolean σ-algebra of \((X, M)\) with a minimal and maximal element \(x( \emptyset)\) and \(x(R^{1})\), respectively. If \(\tau: M \to M\) is a σ-homomorphism and x is an F-observable on \((X,M)\), then it is easy to verify that \(\tau \circ x: E \to \tau (x (E))\), \(E \in B(R^{1})\), is an F-observable on \((X,M)\), too. Let any fuzzy probability space \((X, M, m)\) be given. If x is an F-observable on \((X, M)\), then the mapping \(m_{x}: E \mapsto m(x(E))\), \(E \in B(R^{1})\), is a probability measure on \(B(R^{1})\). The probability that an F-observable x has a value in \(E \in B(R^{1})\) is given by \(m(x(E))\).
We present some examples of the above notions.
Example 2.1
Let \((X, S, P)\) be a classical probability space and \(\xi: X \to R^{1}\) be a random variable in the sense of classical probability theory. Put \(M = \{ \chi_{A}; A \in S \}\), where \(\chi_{A}\) is a characteristic function of a set \(A \in S\), and define the mapping \(m:M \to \langle 0, 1 \rangle\) by \(m(\chi_{A}) = P(A)\). Then the triplet \((X, M, m)\) is a fuzzy probability space and the mapping \(x:B(R^{1}) \to M\) defined by \(x(E) = \chi_{\xi^{ - 1}(E)}\), \(E \in B(R^{1})\), is an F-observable on the fuzzy measurable space \((X, M)\).
Example 2.2
Lemma 2.1
Let x be an F-observable on a fuzzy measurable space \((X, M)\) and \(f: R^{1} \to R^{1}\) be a Borel measurable function. Then the mapping \(f(x):B(R^{1}) \to M\) defined, for every \(E \in B(R^{1})\), by \(f(x)(E) = x(f^{ - 1}(E))\), is an F-observable on \((X, M)\) such that \(R(f(x)) \subset R(x)\).
Proof
In particular, if \(f(t) = t^{2}\), \(t \in R^{1}\), we put \(x^{2}: = f(x)\), etc. If \(k \in R^{1}\), then the mapping \(kx:B(R^{1}) \to M\) defined by \((kx)( E) = x(\{ t \in R^{1}; kt \in E\} )\), for every \(E \in B(R^{1})\), is an F-observable on \((X, M)\).
Since there is a one-to-one correspondence between an F-observable x and the system \(\{ B_{x}(t): = x(( - \infty, t)); t \in R^{1} \}\) (Dvurečenskij and Tirpáková [23]), the sum of any pair of F-observables x, y on \((X, M)\) has been defined in the following way.
Definition 2.4
[23]
By the spectrum of an F-observable x we mean the set \(\sigma (x): =\bigcap\{C \subset R^{1}; C\mbox{ is closed}\mbox{ }\mbox{and }x(C) = x(R^{1})\}\). An F-observable x is called bounded if \(\sigma (x)\) is a bounded set; in this case we define the norm of x via \(\Vert x \Vert = \sup \{ \vert t \vert ; t \in \sigma (x) \}\). Let x and y be two F-observables on \((X,M)\). We write \(x \le y\) if \(\sigma (y - x) \subset \langle 0, \infty )\).
Definition 2.5
[24]
3 Main results
In this section, we give analogies of Mesiar’s ergodic theorems for the case of a fuzzy dynamical system \((X, M, m, \tau)\). In the proofs we will use the method of F-σ-ideals described below as well as the properties of a σ-homomorphism τ. It is noted that these results can be obtained also by factorizing over the σ-ideal of W-empty sets; details of this approach can be found, e.g., in [25].
Let any fuzzy probability space \((X, M, m)\) be given. Put \(I_{ \circ} = \{ a \in M; \exists c \ge 1 / 2 \mbox{ such that}\mbox{ }a \wedge c \le 1 / 2 \}\). It is easy to verify that \(I_{ \circ}\) is a σ-ideal, i.e., (i) if \(a \in M\), \(b \in I_{ \circ}\), \(a \le b\), then \(a \in I_{\circ}\); (ii) if \(\{ a_{i} \}_{i = 1}^{\infty} \subset I_{ \circ}\), then \(\bigvee_{i = 1}^{\infty} a_{i} \in I_{ \circ}\); (iii) \(a \wedge a' \in I_{ \circ}\) for every \(a \in M\); (iv) if \(a \wedge c \in I_{ \circ}\) for some \(c \ge 1 / 2\), \(c \in M\), then \(a \in I_{ \circ}\). In the set M we define the relation of equivalence ∼ in the following way: for every \(a, b \in M\), \(a \sim b\) if and only if \(a \wedge b'\), \(a' \wedge b \in I_{ \circ}\). Put \([a] = \{ b \in M; b \sim a \}\) for every \(a \in M\). The system \(M/I_{ \circ} = \{ [a]; a \in M \}\) is a Boolean σ-algebra, where \(\bigvee_{n = 1}^{\infty} [a_{n}] =[ \bigvee_{n = 1}^{\infty} a_{n}]\) and \([a]' = [a' ]\). It is easy to verify that if \(a_{1}, a_{2} \in [a]\), then \(m(a_{1}) = m(a_{2})\). If we define the mapping \(\mu:M/I_{ \circ} \to \langle 0, 1 \rangle\) by the equality \(\mu ( [a]): = m(a)\) for every \([a] \in M/I_{ \circ}\), then μ is a probability measure on the Boolean σ-algebra \(M/I_{ \circ}\).
Let \((X, M, m, \tau)\) be any fuzzy dynamical system. Then the mapping \(\bar{\tau}:M/I_{ \circ} \to M/I_{ \circ}\) defined by \(\bar{\tau} ([a]) = [\tau (a)]\), \(a \in M\), is a σ-homomorphism of the Boolean σ-algebra \(M/I_{ \circ}\), i.e., for every \(a \in M\), \(\bar{\tau} ([a]') = (\bar{\tau} ([a]))'\), and for every sequence \(\{ a_{n} \}_{n = 1}^{\infty} \subset M\), \(\bar{\tau} ( \bigvee_{n = 1}^{\infty} [a_{n}])= \bigvee_{n = 1}^{\infty} \bar{\tau} ([a_{n}])\); moreover, \(\bar{\tau}\) is invariant in μ, i.e., \(\mu (\bar{\tau} ([a])) = \mu ([a])\), for every \(a \in M\).
Remark 3.1
- (i)
\(\bar{x} (\emptyset) = [0_{X}]\);
- (ii)
\(\bar{x} (E^{C}) = (\bar{x} (E))'\), for every \(E \in B (R^{1})\);
- (iii)
\(\bar{x} ( \bigcup_{n = 1}^{\infty} E_{n}) = \bigvee_{n = 1}^{\infty} \bar{x} (E_{n})\), for any sequence \(\{ E_{n} \}_{n = 1}^{\infty} \subset B (R^{1})\).
Lemma 3.1
- (i)
\(\bar{\tau}^{k} \circ \bar{x}_{n} = h \circ \tau^{k} \circ x_{n}\), for \(k = 1, 2,\ldots\) , \(n = 1, 2,\ldots\) .
- (ii)
Let \(\mathcal{A}\) be the minimal Boolean sub-σ-algebra of \(M/I_{ \circ}\) containing all ranges of \(\bar{\tau}^{k} \circ \bar{x}_{n}\), \(k = 1, 2,\ldots\) , \(n = 1, 2,\ldots\) . Then \(\bar{\tau} ( [a]) \in\mathcal{A}\) for any \([a] \in \mathcal{A}\).
Proof
(ii) Put \(\mathcal{A}_{0} = \{ [a] \in \mathcal{A}; \bar{\tau} ([a]) \in\mathcal{A}\}\). Since \(\mathcal{A}_{0} \subset\mathcal{A}\), it is sufficient to show the inclusion \(\mathcal{A}\subset \mathcal{A}_{0}\). We will prove that \(\mathcal{A}_{0}\) is a Boolean σ-algebra. If \([a] \in \mathcal{A}_{0}\), then \(\bar{\tau} ( [a]) \in\mathcal{A}\), and since \(\mathcal{A}\) is a Boolean σ-algebra, \((\bar{\tau} ( [a]))' \in\mathcal{A}\). The equality \(\bar{\tau} ([a]') = (\bar{\tau} ([a]))'\) implies \([a]' \in \mathcal{A}_{0}\). Let \([a_{n}] \in \mathcal{A}_{0}\) for \(n = 1, 2,\ldots\) . Then \(\bar{\tau} ( [a_{n}]) \in\mathcal{A}\) for \(n = 1, 2,\ldots\) , and since \(\mathcal{A}\) is a Boolean σ-algebra, \(\bigvee_{n = 1}^{\infty} \bar{\tau} ([a_{n}]) \in\mathcal{A}\). From the equality \(\bigvee_{n = 1}^{\infty} \bar{\tau} ([a_{n}]) = \bar{\tau} ( \bigvee_{n = 1}^{\infty} [a_{n}] )\) we get \(\bigvee_{n = 1}^{\infty} [a_{n}] \in \mathcal{A}_{0}\). Moreover, \([0_{X}], [ 1_{X}] \in \mathcal{A}_{0}\). Thus \(\mathcal{A}_{0}\) is a Boolean sub-σ-algebra of \([M]\) containing all ranges of \(\bar{\tau}^{k} \circ \bar{x}_{n}\), \(k = 1, 2,\ldots\) , \(n = 1, 2,\ldots\) , and therefore \(\mathcal{A}\subset \mathcal{A}_{0}\). The proof is finished. □
The following theorem is a fuzzy analogy of Theorem 1.1.
Theorem 3.1
Proof
Theorem 3.2
Proof
But the last assertion follows from Theorem 1.2. The proof is finished. □
Corollary 3.1
Proof
We put \(x_{n} = x_{n}^{ +} - x_{n}^{ -}\), where \(x_{n}^{ +} = f^{ +} \circ x_{n}\), \(x_{n}^{ -} = f^{ -} \circ x_{n}\), \(f^{ +} (t) = \max (0, t)\) and \(f^{ -} (t) = - \min (0, t)\), \(t \in R^{1}\). Then \(\vert x_{n} \vert = x_{n}^{ +} + x_{n}^{ -}\). Applying Theorem 3.2 to both sequences \(\{ x_{n}^{ +} \}_{n = 1}^{\infty}\) and \(\{ x_{n}^{ -} \}_{n = 1}^{\infty}\) we get what was claimed. □
4 Conclusions
In this paper we have presented generalizations of some ergodic theorems from the classical ergodic theory to the fuzzy case. In the proofs the method of F-σ-ideals was used.
Declarations
Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions.
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Authors’ Affiliations
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