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The fuzzy analogies of some ergodic theorems
Advances in Difference Equations volume 2015, Article number: 171 (2015)
Abstract
In the paper (Markechová in Fuzzy Sets Syst. 48:351363, 1992), fuzzy dynamical systems have been defined. In this contribution, using the method of Fσideals, we prove analogies of some ergodic theorems for fuzzy dynamical systems.
1 Introduction
In the classical probability theory, which is based on the Kolmogorov axiomatic system [1], a random event is every element of the σalgebra S of subsets of a set X. A probability is a normalized measure defined on the σalgebra S. The notion of a σalgebra 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, socalled 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 Smeasurable 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 minvariant σhomomorphism. 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 Fσideals we proved a fuzzy version of KolmogorovSinai 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]
Let \((X, S, P)\) be a given probability space, \(\xi_{1},\xi_{2},\ldots\) be a sequence of random variables on it. Let \(T:X \to X\) be a measure preserving transformation, \(K > 0\) a real constant. Suppose that \(\vert \xi_{n} \vert \le K\) for \(n = 1,2,\ldots\) and \(\xi_{n} \to 0\) almost everywhere in P (i.e., \(P( \bigcup_{k = 1}^{\infty} \bigcap_{n = k}^{\infty} \xi_{n}^{  1}( \langle  \varepsilon, \varepsilon \rangle )) = 1\)). Then
Theorem 1.2
[17]
Let \((X, S, P)\) be a given probability space, \(\zeta,\xi_{1},\xi_{2},\ldots\) be random variables on it such that \(0 \le \xi_{n} \le \zeta\) for \(n = 1,2,\ldots\) . Let \(T:X \to X\) be a measure preserving transformation and \(\xi_{n} \to 0\) almost everywhere in P. Then
In Section 3 we give fuzzy analogies of the above results. In the proofs we will use the method of Fσideals. 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]
A fuzzy probability space is a triplet \((X,M,m)\), where X is a nonempty set; M is a fuzzy σalgebra of fuzzy subsets of X (i.e., (i) \(1_{X} \in M\); \((1 / 2)_{X} \notin M\); (ii) if \(a_{n} \in M\), \(n = 1, 2,\ldots\) , then \(\bigvee_{n = 1}^{\infty} a_{n} \in M\); (iii) if \(a \in M\), then \(a' \in M\)) and the mapping \(m:M \to \langle 0, \infty ) \) fulfils the following conditions:

(P1)
\(m(a \vee a') = 1\) for every \(a \in M\);

(P2)
if \(\{ a_{n} \}_{n = 1}^{\infty}\) is a sequence of pairwise Wseparated 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 σlattice 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 σalgebra 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 (Wnotions) were defined by Piasecki in [21] as follows: each fuzzy subset \(a \in M\) such that \(a \ge a'\) is called a Wuniversum; each fuzzy subset \(a \in M\) such that \(a \le a'\) is called a Wempty set. Fuzzy subsets \(a, b \in M\) such that \(a \le b'\) are called Wseparated. It can be proved that a fuzzy set \(a \in M\) is a Wuniversum 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 Pmeasure. Any fuzzy Pmeasure 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 minvariant σhomomorphism, 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 Fobservable.
Definition 2.3
[22]
An Fobservable on a fuzzy measurable space \((X, M)\) is a mapping \(x:B(R^{1}) \to M\) such that

(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 \(\{ E_{n} \}_{n = 1}^{\infty} \subset B(R^{1})\),
where \(B(R^{1})\) is the family of all Borel subsets of the real line \(R^{1}\) and \(E^{C}\) denotes the complement of a set \(E \subset R^{1}\).
It is easy to see that if x is an Fobservable, then the range of the Fobservable x, i.e., the set \(R(x): = \{ x(E); E \in B(R^{1})\}\), is a Boolean σalgebra of \((X, M)\) with a minimal and maximal element \(x( \emptyset)\) and \(x(R^{1})\), respectively. If \(\tau: M \to M\) is a σhomomorphism and x is an Fobservable 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 Fobservable on \((X,M)\), too. Let any fuzzy probability space \((X, M, m)\) be given. If x is an Fobservable 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 Fobservable 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 Fobservable on the fuzzy measurable space \((X, M)\).
Example 2.2
Let \((X, M)\) be any fuzzy measurable space. If \(a \in M\), then the mapping \(x_{a}\) defined by putting
for any \(E \in B(R^{1})\), is an Fobservable on the fuzzy measurable space \((X, M)\), called the question observable of the fuzzy set a. It is evident that \(x_{a}\) plays the role of an indicator of the fuzzy set a. We denote the question observable of the empty fuzzy set \(0_{X}\) by σ, i.e., \(\sigma = x_{0_{X}}\). We have
Lemma 2.1
Let x be an Fobservable 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 Fobservable on \((X, M)\) such that \(R(f(x)) \subset R(x)\).
Proof
For every \(E \in B(R^{1})\),
and for any sequence \(\{ E_{n} \}_{n = 1}^{\infty} \subset B(R^{1})\),
This means that the mapping \(f(x):B(R^{1}) \to M\) is an Fobservable on \((X, M)\). If \(a \in R(f(x))\), then \(a = f(x)(E) = x(f^{  1}(E))\) for some set \(E \in B(R^{1})\). Since the function \(f: R^{1} \to R^{1}\) is a Borel measurable function, \(f^{  1}(E) \in B(R^{1})\), and hence \(a \in R(x)\). The proof is finished. □
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 Fobservable on \((X, M)\).
Since there is a onetoone correspondence between an Fobservable x and the system \(\{ B_{x}(t): = x((  \infty, t)); t \in R^{1} \}\) (Dvurečenskij and Tirpáková [23]), the sum of any pair of Fobservables x, y on \((X, M)\) has been defined in the following way.
Definition 2.4
[23]
Let \((X, M)\) be a fuzzy measurable space. By the sum of any pair of two Fobservables x, y on \((X, M)\) we mean an Fobservable z such that
where Q is the set of all rational numbers in the real line \(R^{1}\). We write \(z = x + y\).
In [23], it has been proved that the sum of any pair of Fobservables always exists and it is determined uniquely. Moreover, for any Fobservables x, y, z on \((X, M)\) we have \(x + y = y + x\); \((x + y) + z = x + (y + z)\). The difference of two Fobservables x, y on \((X, M)\) is defined by \(x  y = (x + (  y))\), where \((  y)(E) = y ( \{ t;  t \in E \} )\), for every \(E \in B(R^{1})\). The product \(x \cdot y\) of two Fobservables x, y on \((X, M)\) is defined as follows:
By the spectrum of an Fobservable 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 Fobservable 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 Fobservables on \((X,M)\). We write \(x \le y\) if \(\sigma (y  x) \subset \langle 0, \infty )\).
Definition 2.5
[24]
Let \((X, M, m)\) be a given fuzzy probability space, \(x, x_{1}, x_{2},\ldots\) be Fobservables on \((X, M)\). We say that the sequence \(\{ x_{n} \}_{n = 1}^{\infty}\) converges to x almost everywhere in m (and we write \(x_{n} \to x\) ma.e.), if, for every \(\varepsilon > 0\),
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 Fσideals described below as well as the properties of a σhomomorphism τ. It is noted that these results can be obtained also by factorizing over the σideal of Wempty 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 σideal, 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 σalgebra, 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 σalgebra \(M/I_{ \circ}\).
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 σhomomorphism of the Boolean σalgebra \(M/I_{ \circ}\), 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
Let \((X, M)\) be a given fuzzy measurable space and x be an Fobservable on it. Let us define the mapping \(h: M \to M/I_{ \circ}\) via \(h(a) = [a ]\), \(a \in M\). Then it is easy to verify that h is a σhomomorphism from M onto \(M/I_{ \circ}\) and \(\bar{x}: = h \circ x\) is an observable on the Boolean σalgebra \(M/I_{ \circ}\), i.e., the following properties hold:

(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
Let \((X, M, m, \tau)\) be a fuzzy dynamical system and let \(\{ x_{n} \}_{n = 1}^{\infty}\) be a sequence of Fobservables on \((X,M)\). If h is the mapping from the preceding remark and \(\bar{x}_{n}: = h \circ x_{n}\), \(n = 1, 2,\ldots\) , then we have:

(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 subσalgebra 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
(i) For every \(E \in B(R^{1})\) we have
i.e., \(\bar{\tau}^{k} \circ \bar{x}_{n} = h \circ \tau^{k} \circ x_{n}\), \(n = 1, 2,\ldots\) , \(k = 1,2,\ldots\) .
(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 σalgebra. If \([a] \in \mathcal{A}_{0}\), then \(\bar{\tau} ( [a]) \in\mathcal{A}\), and since \(\mathcal{A}\) is a Boolean σalgebra, \((\bar{\tau} ( [a]))' \in\mathcal{A}\). 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 σalgebra, \(\bigvee_{n = 1}^{\infty} \bar{\tau} ([a_{n}]) \in\mathcal{A}\). 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 subσalgebra 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. □
Figure 1 shows the basic idea of using the method of Fσideals for the verification of extending the possibility of ergodic theory on a fuzzy measurable space.
The following theorem is a fuzzy analogy of Theorem 1.1.
Theorem 3.1
Let \((X,M,m,\tau)\) be a given fuzzy dynamical system and \(\{ x_{n} \}_{n = 1}^{\infty}\) be a sequence of bounded Fobservables on \((X,M)\). Let there exists a real constant \(K > 0\) such that \(\Vert x_{n} \Vert \le K\) for \(n = 1, 2,\ldots\) . Suppose \(x_{n} \to \sigma\) almost everywhere in m. Then
Proof
The Boolean subσalgebra \(\mathcal{A}\) in Lemma 3.1 has a countable generator, hence, due to Varadarajan [26], there exists an observable
and a sequence of realvalued Borel functions \(\{ f_{n} \}_{n = 1}^{\infty}\), such that
The sequence \(\{ f_{n} \}_{n = 1}^{\infty}\) is essentially unique in the following sense: if \(z (f_{n}^{  1}(E)) = z (g_{n}^{  1}(E))\), \(E \in B (R^{1})\), then \(z ( \{ t: f_{n}(t ) \ne g_{n}(t ) \} ) = [0_{X}]\). From the construction of z it follows that \(\bar{\tau}\) is zmeasurable, i.e., \(\bar{\tau} (z (B (R^{1}) )) \subset z (B (R^{1}))\). Due to Dvurečenskij and Riečan [22], this is possible iff there is a Borel measurable transformation \(T: R^{1} \to R^{1}\) such that
Therefore we have
and consequently, for every \(E \in B (R^{1})\),
Moreover, \(\sigma (x_{n}) \supset \sigma (\bar{x}_{n}) \supset \sigma (f_{n}^{  1})\), and therefore \(\vert f_{n} \vert \le \Vert \bar{x}_{n} \Vert \le \Vert x_{n} \Vert \le K\).
Take into account the system \((R^{1}, B (R^{1}), \mu_{z}, T)\), where \(\mu_{z}\) is the mapping defined by \(\mu_{z} (E) = \mu (z (E))\), for every \(E \in B (R^{1})\). Then \(\mu_{z}\) is a probability measure on \(B (R^{1})\). Moreover, for every \(E \in B (R^{1})\),
i.e., the transformation T is \(\mu_{z}\)invariant. This means that the system \((R^{1},B (R^{1}),\mu_{z},T)\) is a dynamical system in the classical sense. For observables in a Boolean σalgebra, there is a wellknown way to define their sum [24], and the convergence almost everywhere of observables in \(M/I_{ \circ}\) is the same as for Fobservables.
By the assumption \(x_{n} \to \sigma\) almost everywhere in m, i.e., for every \(\varepsilon > 0\),
But
Hence, due to Theorem 1.1, we have
On the other hand,
where \(\bar{\sigma} (E) = [0_{X}]\) if \(0 \notin E\) and \(\bar{\sigma} (E) = [1_{X}]\) in the other cases. The previous convergence is true if and only if
which is possible if and only if
Therefore, (3.1) is proved. □
Theorem 3.2
Let \((X,M,m,\tau)\) be any fuzzy dynamical system and \(\{ x_{n} \}_{n = 1}^{\infty}\) be a sequence of Fobservables on \((X,M)\). Let y be an Fobservable on \((X,M)\) such that \(\sigma \le x_{n} \le y\) for \(n = 1,2,\ldots\) . Suppose \(x_{n} \to \sigma\) almost everywhere in m. Then
Proof
We will use similar arguments to above. Let \(\mathcal{A}_{1}\) be the minimal Boolean subσalgebra of \(M/I_{ \circ}\) containing all ranges of \(\bar{\tau}^{k} \circ \bar{x}_{n}\) and \(\bar{\tau}^{k} \circ \bar{y}\) for \(k = 1, 2,\ldots\) , \(n = 1, 2,\ldots\) . Then \(\bar{\tau} \mathcal{A}_{1} \subset \mathcal{A}_{1}\) and \(\mathcal{A}_{1}\) has a countable generator. In view of Varadarajan [26], there is an observable \(z: B(R^{1}) \to M/I_{ \circ}\) such that
Moreover, there are realvalued Borel functions f, \(f_{n}\), \(n = 1, 2,\ldots\) , such that
Denote \(g_{n} = \max (0,f_{n})\), \(n = 1, 2,\ldots\) , and \(g = \max (0,f)\). Then, for every \(E \in B (R^{1})\), we have \(\bar{x}_{n} (E) = z (g_{n}^{  1}(E))\), \(n = 1, 2,\ldots\) , and \(\bar{y}(E) = z(g^{  1}(E))\). Let \(h_{n} = \min (f_{n},f)\), \(n = 1, 2,\ldots\) , and \(h = f\). Then \(\bar{x}_{n} (E) = z (h_{n}^{  1}(E))\), \(\bar{y}(E) = z(h^{  1}(E))\). Moreover, \(0 \le \Vert \bar{x}_{n} \Vert \le \Vert \bar{y} \Vert \) and \(0 \le h_{n} \le h\). Similar to the proof of the preceding theorem, we take into account the dynamical system \((R^{1}, B (R^{1}), \mu_{z}, T)\), where \(\mu_{z}: E \to \mu (z (E))\), \(E \in B (R^{1})\), is a probability measure on \(B (R^{1})\) such that \(\mu_{z} (T^{  1}(E)) = \mu_{z} (E)\), for every \(E \in B (R^{1})\), and \(\bar{\tau} (z(E)) = z (T^{  1}(E))\), \(E \in B (R^{1})\). Thus we have
But the last assertion follows from Theorem 1.2. The proof is finished. □
Corollary 3.1
Let \((X,M,m,\tau)\) be any fuzzy dynamical system and \(\{ x_{n} \}_{n = 1}^{\infty}\) be a sequence of Fobservables on \((X,M)\). Let y be an Fobservable on \((X,M)\) such that \(\vert x_{n} \vert \le y\) for \(n = 1,2,\ldots \) . If \(x_{n} \to \sigma\) almost everywhere in m, then
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 Fσideals was used.
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Tirpáková, A., Markechová, D. The fuzzy analogies of some ergodic theorems. Adv Differ Equ 2015, 171 (2015). https://doi.org/10.1186/s1366201504992
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DOI: https://doi.org/10.1186/s1366201504992