Conditional logical entropy of fuzzy σ -algebras

1 Introduction

Entropy is a tool to measure the amount of uncertainty in random events. The entropy has been applied in the information theory, physics, computer sciences, statistics, chemistry, biology, sociology, general systems theory and many other fields. The classical approach in the information theory was based on Shannon entropy [8, 20]. Shannon entropy of a probability distribution was studied in [17]. Kolmogorov and Sinai used the Shannon entropy to define the entropy of measurable partitions and then they defined the entropy of dynamical systems. Kolmogorov - Sinai entropy is a useful tool in studying the isomorphism of dynamical systems. In fact, the isomorphic dynamical systems have the same entropy [3, 21]. Thus, they developed a method for distinguishing non-isomorphic dynamical systems.

In [9, 19], Good, Patil, Taillie and Rao defined and studied the concept of the logical entropy. Let P = (p₁,. . . , p _n ) be a probability distribution. The logical entropy of P is defined in [6] by the formula h ( P ) = ∑ i = 1 n p i ( 1 - p i ) . Rao introduced precisely this concept as a quadratic entropy [19]. In the years 2009 and 2013, this concept was discussed by Ellerman [6, 7] and the author [7] investigated the relationship between the logical entropy and Shannon entropy. Ebrahimzadeh, Eslami, Markechova and Riecan defined and studied the notion of logical entropy of partitions and dynamical systems on some algebraic structures [2, 16].

Markechova [14, 15] studied Shannon entropy of complete partitions and entropy of an F-dynamical system. Fuzzy partition theory in a different way using of triangular norms and its entropy were developed by Dumitrescu [1] (see also [10]). Klement in [13] defined the notion of F-probability measure space by defining the notions of fuzzyσ-algebra and F-measure. Mona Khare introduced the notions of Shannon entropy and conditional entropy of fuzzy σ-algebras having finitely many atoms on anF-probability measure space and presented some results concerning this measures [11, 22].

The main idea of the present research is to study analogues of these results for the cases of logical entropy and logical mutual information onF-probability measure spaces. We show that the basic properties of Shannon entropy of fuzzy σ-algebras on F-probability measure spaces, valid for the case of the logical entropy, too. So the suggested measure can be used (in addition to the Shannon entropy of fuzzy σ-algebras) as a measure of information of experiments whose outcomes are fuzzy events.

In Section 2, some basic definitions are presented that will be useful in further considerations. In Section 3, the logical entropy and conditional logical entropy of fuzzy sub σ-algebras on the F-probability measure space are defined and basic properties of these measures are investigated. In the subsequent section the logical entropy and conditional logical entropy under the relation of equivalence modulo 0, is studied. In Section 4, the logical mutual information of fuzzy sub σ-algebras on the F-probability measure space is defined and some properties of this measure are found.

2 Basic notions

In this section, some basic definitions are presented that will be useful in further considerations.

Definition 2.1. [13] A fuzzy σ-algebra on a nonempty set X is a subfamily of I ^X which satisfies the following conditions:

1 ∈ M ,

If N 1 and N 2 are fuzzy σ-algebras on X, then N 1 ∨ N 2 is the smallest fuzzy σ-algebra on X containing N 1 ∪ N 2 [13].

Definition 2.2. [13] An F-probability measure m on M is a function m : M ⟶ I which satisfies:

m (1) =1,

The triple ( X , M , m ) is called an F-probability measure space.

Example 2.3. Let X = [0, 1] and let M = { λ 1 , λ 2 , λ 3 , λ 4 } where λ _i  : [0, 1] → [0, 1] defined by λ₁ (x) =1, λ₂ (x) =0, λ 3 ( x ) = { 1 if x ∈ [ 0 , 1 2 ] 0 if x ∈ [ 1 2 , 1 ] and λ 4 ( x ) = { 0 if x ∈ [ 0 , 1 2 ] 1 if x ∈ [ 1 2 , 1 ] .

It is clear that M is a fuzzy σ-algebra on X . Define m : M → [ 0 , 1 ] by m ( λ i ) = ∫ 0 1 λ i ( x ) d x .

Also let ∨ and ∧ mean respectively supremum and infimum. It is easy to see that m is an F-probability measure on M . So ( X , M , m ) is an F-probability measure space.

Definition 2.4. [23] Let ( X , M , m ) be an F-probability measure space.

For λ , μ ∈ M , a relation = (mod m) is defined as λ = μ ( mod m ) ⇔ m ( λ ) = m ( μ ) = m ( λ ∨ μ ) .

The relation = (mod m) is an equivalence relation on M . The set of all equivalence classes induced by this relation is denoted by M ˜ and μ ˜ denotes the equivalence class determined by μ.

Elements λ , μ ∈ M are called m-disjoint if λ ∧ μ = 0 (mod m), i.e. m ( λ ∧ μ ) = 0 .

Definition 2.5. [23] Let ( X , M , m ) be an F-probability measure space, and N be a fuzzy sub-σ-algebra of M ˜ . An element μ ˜ ∈ N ˜ is called an atom of N if m (μ) >0, for any λ ˜ ∈ N ˜ , m ( λ ∧ μ ) = m ( λ ) ≠ m ( μ ) ⇒ m ( λ ) = 0 .

The set of all atoms of N is denoted by N ¯ , and F ( M ) denotes the collection of fuzzy sub-σ-algebras of M having finitely many atoms.

Example 2.6. Consider Example 2.3. We have m (λ₁) =1, m (λ₂) =0 and m ( λ 3 ) = m ( λ 4 ) = 1 2 . By Definition 2.4, we obtain ∀i ≠ j, [λ _i ] ≠ [λ _j ] and therefore M ˜ = { λ 1 ˜ , λ 2 ˜ , λ 3 ˜ , λ 4 ˜ } . Suppose N = M ˜ . From Definition 2.5, by a simple calculating we have N ¯ = { λ 3 ˜ , λ 4 ˜ } .

Definition 2.7. [23] Let ( X , M , m ) be an F-probability measure space, and N 1 , N 2 be fuzzy sub-σ-algebras of M . Then N 2 is called an m-refinement of N 1 , written as N 1 ≤ m N 2 , if for μ ∈ N 2 ¯ there exists λ ∈ N 1 ¯ such that m (λ ∧ μ) = m (μ).

The fuzzy sub -σ-algebras N 1 , N 2 are called m-quivalent, denoted by N 1 ≅ m N 2 , if m ( λ ∧ ( ⋁ { μ : μ ∈ N 2 ¯ } ) ) = m ( λ ) , for each λ ∈ N 1 ¯ , and m ( μ ∧ ( ⋁ { λ : λ ∈ N 1 ¯ } ) ) = m ( μ ) , for each μ ∈ N 2 ¯ .

The relation of “m-equivalence” is an equivalence relation on F ( M ) , and [ N ] denotes the set of all m-equivalent fuzzy sub -σ-algebras in F ( M ) [12].

Theorem 2.8. [23] Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 , N 3 be elements of F ( M ) . If N 1 ≤ m N 2 , then N 1 ∨ N 3 ≤ m N 2 ∨ N 3 .

Definition 2.9. [22] For N 1 , N 2 ∈ F ( M ) , the relation ∼ is defined as follows: N 1 ∼ N 2 ⇔ N 1 ≤ m N 2 and N 2 ≤ m N 1 .

Then ∼ is an equivalence relation on F ( M ) and this relation is called equivalence modulo 0 [22].

3 Logical entropy and conditional logical entropy

In this section, we define the notions of logical entropy and conditional logical entropy of fuzzy σ-algebras on an F-probability measure space. We prove some ergodic properties of the suggested measures. For example, we study the conditional logical entropy under the common refinement of fuzzy σ-algebras and prove the subadditivity property of logical entropy of fuzzy σ-algebras on an F-probability measure space. We show that the logical entropy and logical conditional entropy of fuzzy σ-algebras of an F-probability measure space are invariant under the relation of equivalence modulo 0.

Definition 3.1. Let ( X , M , m ) be an F-probability measure space and N ∈ F ( M ) and let N ¯ = { λ i : i = 1 , ⋯ , s } . We define the logical entropy h l ( N ) by: h l ( N ) : = ∑ i = 1 s m ( λ i ) ( 1 - m ( λ i ) ) .

Example 3.2. Consider Examples 2.3 and 2.6. We obtained N ¯ = { λ 3 , λ 4 } . Since m ( λ 3 ) = m ( λ 4 ) = 1 2 , by Definition 3.1 we have h l ( N ) = 1 2 .

Definition 3.3. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 ∈ F ( M ) , N 1 ¯ = { λ i : i = 1 , ⋯ , s } and N 2 ¯ = { μ i : j = 1 , ⋯ , t } . The logical conditional entropy h l ( N 1 / N 2 ) is defined as: h l ( N 1 / N 2 ) : = ∑ i = 1 s ∑ j = 1 t m ( λ i ∧ μ j ) ( m ( μ j ) - m ( λ i ∧ μ j ) ) .

Note that h l ( N 1 / N 2 ) ≥ 0 and h l ( N 1 / N 1 ) = 0 .

Now the assertion of the following lemma will be proved that will be useful in the next theorem.

Lemma 3.4. Let ( X , M , m ) be an F-probability measure space, N 1 , N 2 , N 3 ∈ F ( M ) , and let N 1 ¯ = { λ i : i = 1 , ⋯ , s } , N 2 ¯ = { μ j : j = 1 , ⋯ , t } and N 3 ¯ = { ν k : k = 1 , ⋯ , p } . If N 1 ≅ m N 2 ≅ m N 3 , then

∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) m ( ν k ) = ∑ i = 1 s ∑ k = 1 p m ( λ i ∧ ν k ) m ( ν k ) ,

Proof. It is proved in [11] that N 1 ∨ N 3 ≅ m N 2 . Hence we have ∑ j = 1 t m ( λ i ∧ μ j ∧ ν k ) = m ( ⋁ j = 1 s ( λ i ∧ μ j ∧ ν k ) ) = m ( ( λ i ∧ ν k ) ∧ ( ⋁ j = 1 t μ j ) ) = m ( λ i ∧ ν k ) .

So we obtain the assertion.

ii) Since N 1 ≅ N 2 we have for each j ∈ {1, ⋯ , t} ,

∑ i = 1 s m ( λ i ∧ μ j ) = m ( ⋁ i = 1 s ( λ i ∧ μ j ) ) = m ( ⋁ i = 1 s λ i ∧ μ j ) = m ( μ j ) .

Thus ∑ i = 1 s ∑ j = 1 t m ( λ i ∧ μ j ) m ( μ j ) = ∑ j = 1 t m ( μ j ) ∑ i = 1 s m ( λ i ∧ μ j ) = ∑ j = 1 t ( m ( μ j ) ) 2 . □

In the following theorem we study the conditional logical entropy under the common refinement of fuzzy σ-algebras.

Theorem 3.5 Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 , N 3 be elements of F ( M ) . If N 1 ≅ m N 2 ≅ m N 3 , then h l ( N 1 ∨ N 2 / N 3 ) = h l ( N 1 / N 3 ) + h l ( N 2 / N 1 ∨ N 3 ) .

Proof. Let N 1 ¯ = { λ i : i = 1 , ⋯ , s } , N 2 ¯ = { μ j : j = 1 , ⋯ , t } and N 3 ¯ = { ν k : k = 1 , ⋯ , p } . From Lemma 3.4, we have h l ( N 1 ∨ N 2 / N 3 ) = ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) ( m ( ν k ) - m ( λ i ∧ μ j ∧ ν k ) ) = ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) m ( ν k ) - ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p ( m ( λ i ∧ μ j ∧ ν k ) ) 2 = ∑ i = 1 n ∑ k = 1 m m ( λ i ∧ ν k ) m ( ν k ) - ∑ i = 1 n ∑ k = 1 m ( m ( λ i ∧ ν k ) ) 2 + ∑ i = 1 s ∑ k = 1 p ( m ( λ i ∧ ν k ) ) 2 - ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p ( m ( λ i ∧ μ j ∧ ν k ) ) 2 .

On the other hand, we have h l ( N 1 / N 3 ) = ∑ i = 1 s ∑ k = 1 p m ( λ i ∧ ν k ) ( m ( ν k ) - m ( λ i ∧ ν k ) ) = ∑ i = 1 s ∑ k = 1 p m ( λ i ∧ ν k ) m ( ν k ) - ∑ i = 1 s ∑ k = 1 p ( m ( λ i ∧ ν k ) ) 2 , and h l ( N 2 / N 1 ∨ N 3 ) = ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) ( m ( λ i ∧ ν k ) - m ( λ i ∧ μ j ∧ ν k ) ) = ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) m ( λ i ∧ ν k ) - ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p ( m ( λ i ∧ μ j ∧ ν k ) ) 2 = ∑ i = 1 s ∑ k = 1 p ( m ( λ i ∧ ν k ) ) 2 - ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p ( m ( λ i ∧ μ j ∧ ν k ) ) 2 .

Combining the above relations we obtain the assertion. □

In the next theorem the subadditivity property of logical entropy of fuzzy σ-algebras on anF-probability measure space is proved.

Theorem 3.6. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 be elements of F ( M ) . If N 1 ≅ m N 2 , then

h l ( N 1 / { 0 , 1 } ) = h l ( N 1 ) ,

Proof. Let N 1 ¯ = { λ i : i = 1 , ⋯ , s } and N 2 ¯ = { μ j : j = 1 , ⋯ , t } .

By Definitions 2.1 iii) and 2.2 iii) we have for each i, m ( λ i ∧ 1 ) = m ( λ i ) + m ( 1 ) - m ( λ i ∨ 1 ) = m ( λ i ) .

So h l ( N 1 / { 1 } ) = ∑ i = 1 s m ( λ i ∧ 1 ) ( m ( 1 ) - m ( λ i ∧ 1 ) ) = h l ( N 1 ) .

Theorem 3.7. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 , N 3 be elements of F ( M ) . If N 1 ≅ m N 2 ≅ m N 3 , then

h l ( N 1 / N 2 ∨ N 3 ) ≤ h l ( N 1 / N 2 ) ,

Proof. i) Let N 1 ¯ = { λ i : i = 1 , ⋯ , s } , N 2 ¯ = { μ j : j = 1 , ⋯ , t } and N 3 ¯ = { ν k : k = 1 , ⋯ , p } . Since N 1 ∨ N 2 ≅ m N 3 , from [22] we have

m ( λ i ∧ μ j ) = m ( ( λ i ∧ μ j ) ∧ ⋁ k = 1 p ν k ) = m ( λ i ∧ ⋁ k = 1 p ( μ j ∧ ν k ) ) = m ( λ i ∧ ⋁ k = 1 p ( η jk ) ) = ∑ k = 1 p m ( λ i ∧ η jk ) . where η _jk  = μ _j  ∧ ν _k , 1 ≤ j ≤ t, 1 ≤ k ≤ p.Thus h l ( N 1 / N 2 ) = ∑ i = 1 s ∑ j = 1 t m ( λ i ∧ μ j ) ( m ( μ j ) - m ( λ i ∧ μ j ) ) = ∑ i = 1 s ∑ j = 1 t ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) ( ∑ k = 1 p m ( μ j ∧ ν k ) - ∑ k = 1 p m ( λ i ∧ μ j ∧ ν k ) ) ≥ ∑ i = 1 s ∑ j = 1 t ∑ k = 1 m m ( λ i ∧ μ j ∧ ν k ) ( m ( μ j ∧ ν k ) - m ( λ i ∧ μ j ∧ ν k ) ) = h l ( N 1 / N 2 ∨ N 3 ) .

ii) The desired result follows from the first part of this theorem and Theorem 3.6. □

The next theorem shows that the logical entropy and logical conditional entropy of fuzzy σ-algebras of an F-probability measure space are invariant under the relation of equivalence modulo 0.

Theorem 3.8. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 , N 3 ∈ F ( M ) . If N 1 ≅ m N 2 ≅ m N 3 , then

h l ( N 1 | N 2 ) = 0 ⇔ N 1 ≤ m N 2 ,

Proof. i) Let N 1 ¯ = { λ i : i = 1 , ⋯ , s } , N 2 ¯ = { μ j : j = 1 , ⋯ , t } and N 1 ≤ m N 2 . Then, for any μ j ∈ N 2 , there exists λ i ∈ N 1 such that m (λ _i  ∧ μ _j ) = m (μ _j ). This means m (λ _i  ∧ μ _j ) - m (μ _j ) =0 and so h l ( N 1 | N 2 ) = 0 . Conversely, let h l ( N 1 | N 2 ) = 0 . Since m (λ _i  ∧ μ _j ) ≥0 and m (λ _i  ∧ μ _j ) ≤ m (μ _j ) we get for each i, j, m (λ _i  ∧ μ _j ) = m (μ _j ) orm (λ _i  ∧ μ _j ) =0 . Now by the proof of Proposition 4.7 in [22] we obtain N 1 ≤ m N 2 .

ii) This follows from part i) of this theorem and Theorem 3.6.

iii) Since N 1 ≤ m N 2 , from the first part of this theorem we have h l ( N 1 | N 2 ) = h l ( N 1 ∨ N 2 ) - h l ( N 2 ) = 0 , therefore h l ( N 1 ∨ N 2 ) = h l ( N 2 ) . Similarly N 2 ≤ m N 1 , implies h l ( N 2 ∨ N 1 ) = h l ( N 1 ) . Since N 1 ∨ N 2 = N 2 ∨ N 1 we conclude h l ( N 1 ) = h l ( N 2 ) .

iv) Since N 1 ∼ N 2 , by Theorem 2.8, N 1 ∨ N 3 ∼ N 2 ∨ N 3 . Based on part ii) of this theorem and Theorem 3.6 ii) we obtain h l ( N 1 | N 3 ) = h l ( N 1 ∨ N 3 ) - h l ( N 3 ) = h l ( N 2 ∨ N 3 ) - h l ( N 3 ) = h l ( N 2 | N 3 ) .

v) N 2 ∼ N 3 implies N 1 ∨ N 2 ∼ N 1 ∨ N 3 . According to part ii) of this theorem and Theorem 3.6 ii) we get h l ( N 1 | N 2 ) = h l ( N 1 ∨ N 2 ) - h l ( N 2 ) = h l ( N 1 ∨ N 3 ) - h l ( N 3 ) = h l ( N 1 | N 3 ) . □

If m and n are F-probability measures on a fuzzy σ- algebra M , then for p ∈ [0, 1] , pm + (1 - p) n is an F-probability measure on M , too [12].

Theorem 3.9. Let ( X , M , m ) and ( X , M , n ) be two F-probability measure spaces. If N ∈ F ( M ) and p ∈ [0, 1] , then ph l ( N , m ) + ( 1 - p ) h l ( N , n ) ≤ h l ( N , pm + ( 1 - p ) n ) .

Proof. Let N = { λ i , i = 1 , ⋯ s } . Since ∑ i = 1 s ( m ( λ i ) ) 2 + ∑ i = 1 s ( n ( λ i ) ) 2 ≥ 2 ∑ i = 1 s m ( λ i ) n ( λ i ) , we have h l ( N , pm + ( 1 - p ) n ) = ∑ i = 1 s ( pm + ( 1 - p ) n ) ( λ i ) - ∑ i = 1 s ( pm + ( 1 - p ) n ) 2 ( λ i ) = p ∑ i = 1 s m ( λ i ) - p 2 ∑ i = 1 s ( m ( λ i ) ) 2 + ( 1 - p ) ∑ i = 1 s n ( λ i ) - ( 1 - p ) 2 ∑ i = 1 s ( n ( λ i ) ) 2 - 2 p ( 1 - p ) ∑ i = 1 s m ( λ i ) n ( λ i ) ≥ p ∑ i = 1 s m ( λ i ) - p 2 ∑ i = 1 s ( m ( λ i ) ) 2 + ( 1 - p ) ∑ i = 1 s n ( λ i ) - ( 1 - p ) 2 ∑ i = 1 s ( n ( λ i ) ) 2 - p ( 1 - p ) ∑ i = 1 s ( m ( λ i ) ) 2 + ( n ( λ i ) ) 2 ) = p ∑ i = 1 s m ( λ i ) + ( - p 2 - p ( 1 - p ) ) ∑ i = 1 s ( m ( λ i ) ) 2 + ( 1 - p ) ∑ i = 1 s n ( λ i ) + ( - ( 1 - p ) 2 - p ( 1 - p ) ) ∑ ( n ( λ i ) ) 2 = p ∑ i = 1 s m ( λ i ) - p ∑ i = 1 s ( m ( λ i ) ) 2 + ( 1 - p ) ∑ i = 1 s n ( λ i ) - ( 1 - p ) ∑ i = 1 s ( n ( λ i ) ) 2 = ph l ( N , m ) + ( 1 - p ) h l ( N , n ) . □

4 Logical mutual information

In this section, the notion of logical mutual information on an F-probability measure space will be defined and some important results relating to this measure will be proved.

Definition 4.1. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 ∈ F ( M ) , N 1 ¯ = { μ i : i = 1 , ⋯ , s } and N 2 ¯ = { λ j : j = 1 , ⋯ , t } . The mutual information I l ( N 1 , N 2 ) about a fuzzy σ-algebra N 2 in a fuzzy σ-algebras N 1 is defined as: I l ( N 1 , N 2 ) : = ∑ i = 1 s ∑ j = 1 t m ( λ i ∧ μ j ) ( 1 - m ( λ i ) - m ( μ j ) + m ( λ i ∧ μ j ) ) .

Note that in the previous definition since for each i, j, m (λ _i  ∧ μ _j ) = m (μ _i  ∧ λ _j ) we may obtain I l ( N 1 , N 2 ) = I l ( N 2 , N 1 ) .

Theorem 4.2. Let ( X , M , m ) be an F-probability measure space and N 1 , N 2 ∈ F ( M ) . If N 1 ≅ m N 2 ≅ m N 3 , then

I l ( N 1 , N 2 ) = h l ( N 1 ) + h l ( N 2 ) - h l ( N 1 ∨ N 2 ) ,

Proof. Let N 1 ¯ = { λ i : i = 1 , ⋯ , s } and N 2 ¯ = { μ j : j = 1 , ⋯ , t } .

By Definition 4.1, we have I l ( N 1 , N 2 ) = ∑ i = 1 s ∑ j = 1 t m ( λ i ∧ μ j ) [ 1 - m ( λ i ) + ( 1 - m ( μ j ) ) - ( 1 - m ( λ i ∧ μ j ) ] = ∑ i = 1 s m ( λ i ) ( 1 - m ( λ i ) ) + ∑ j = 1 k m ( μ j ) ( 1 - m ( μ j ) ) - ∑ i = 1 n ∑ j = 1 t m ( λ i ∧ μ j ) ( 1 - m ( λ i ∧ μ j ) ) = h l ( N 1 ) + h l ( N 2 ) - h l ( N 1 ∨ N 2 ) .

According to Theorem 3.6 iv) and part i) of the previous theorem we obtain I _l  (N₁, N₂) ≥0 . Since h l ( N 1 | N 1 ) = 0 , part ii) of the previous theorem implies that I l ( N 1 , N 1 ) = h l ( N 1 ) .

5 Conclusion

This work has introduced the notions of logical entropy and logical conditional entropy of fuzzyσ-algebras on F-probability measure spaces. It was shown that the chain rules for logical entropy of fuzzy σ-algebras in F-probability measure spaces are established. The property of subadditivity for the logical entropy of fuzzy σ-algebras was proved. Furthermore, the notion of logical mutual information on F-probability measure spaces was studied. It was shown that the basic properties of Shannon entropy of fuzzy σ-algebras on F-probability measure spaces, valid for the case of the logical entropy, too. So the suggested measure can be used (in addition to the Shannon entropy of fuzzy σ-algebras) as a measure of information of experiments whose outcomes are fuzzy events.

The aim of our further research is to define using of the concept of logical entropy of fuzzy σ-algebras, the notion of logical entropy of dynamical systems on F-probability measure spaces. Then the notion of logical entropy of dynamical systems can be a new tool for distinction of non-isomorphic fuzzy dynamical systems.