Conditional measures on MV-algebras

Conditional measures on MV-algebras 31 CONDITIONAL MEASURES ON MV-ALGEBRAS Martin Kalina, Oga Nánásiová Dept. of Mathematic, Slovak Univ. of Technology Radlinského 11, 813 68 Bratislava, Slovakia [email protected] , [email protected] Abstract In recent years many papers have been written generalizing some theorems, known from the Kolmogorovian probability theory, to MV-algebras. To achieve such results, so-called product MV-algebras were introduced and, using the product, the joint probability distribution was defined. In this paper we present an approach how to define the joint distributions on MV-algebras which are not necessarily closed under product. First we construct conditional measures on a given MV-algebra. And using these conditional measures we define the joint probability distributions. 1. PRELIMINARIES Definition 1.1. An MV-algebra is 5-tuple )1,0,,,M( / ∗ ⊕ such that ),,( 0/⊕M is an Abelian monoid and moreover • xx = ∗∗ • 10 =/ ∗ • 11 = ⊕ x • xxyyyx ⊕⊕=⊕⊕ ∗∗∗∗ )()( Moreover for all M, ∈ yx we can define ∗∗∗ ⊕=⊗ )( yxyx yyxyx ⊗⊕=∧ ∗ )( y)yx(yx ⊕⊗=∨ ∗ and then )1,0,,(M, / ∧ ∨ is a bounded distributive lattice. Example 1.1. ([2], [13]) Denote ),( ℑ Ω a measurable space and µ some normed measure on that space. Then if we put Μ a system of ]1,0[ -valued ℑ -measurable functions, closed under the following operations ))()(,1min()( xgxfxgf + = ⊕ )1)()(,0max()( − + = ⊗ xgxfxgf )(1)( xfxf −= ∗ , and such that Μ ∈0, then Μ is an MV-algebra. In this paper we will deal just with the MV-algebra Μ from Example 1.1. Denote  = µν df)f( , then ν is an additive measure on Μ , i.e. )g()f()gf( ν ν ν + = ⊕ (1) if )1( gf − ≤ Definition 1.2. Events Μ ∈ gf , will be called ν - orthogonal (for short just orthogonal) if 0)( = ∧ gf ν . 2. CONSTRUCTION OF CONDITIONAL MEASURES ON MV-ALGEBRAS Definition 2.1. For any Μ ∈ g such that 0)( > g ν we say that ).( g γ is a conditional measure if and only if )g.( γ and ).( ∗ g γ are non-negative functions on Μ , bounded from above by 1, such that A. )gf()g()gf()g()f( ∗∗ += γνγνν , for any Μ ∈ f B. 0=)hf( γ , for all Μ ∈ h,f , which are orthogonal to each other Definition 2.1. immediately implies the following Lemma 2.1. For any Μ ∈ g such that 0)( > g ν there holds 1)1( =g γ Definition 2.2. Denote Τ the system of all transformations Ω Μτ ],[: 10→ such that for each Μ ∈ f 1. τ is ℑ -measurable 2.   = µτµ dfdf )( 3. for any Ω ∈ x 0)( = xf ⇔ .0))(( = xf τ Theorem 2.2. Let Ττ ∈ 2 be such that for any Μ ∈ g )(1)( 22 gg ττ −= ∗ and Ττ ∈ 1 be an arbitrary transformation. Define for any Μ ∉ g,f          << = = =   10 1 00 2 21 )g( d)g( d)g()f( )g()f( )g( )gf( ν µτ µττ νν ν γ (2) Advances in Electrical and Electronic Engineering 32 Then for any Μ ∈ g such that 0)( > g ν , )g.( γ is a conditional measure. Throughout this paper we will always denote by )( gf γ the measure given by Formula (2). Definition 2.3. We will say that event fis independent of g with respect to a conditional measure γ if and only if )()( gff γν =. In the sequel γ will always denote the conditional measure defined by Formula (2) from Theorem 2.2. The independence of event f of g does not imply the independence of the event g of the event f. This nonsymetric relation of independence allows us to distinguish between a cause and its effects. Similar results concerning the ortho-modular lattices have been achieved also by O. Nánásiová in [5]-[8]. Once having defined for any pair gf , of elements of the MV-algebra Μ the measure )( gf γ , which is the conditional measure if 0)( > g ν , we can define also the two-dimensional joint distribution on Μ × Μ - the measure (probability) of occurence of this pair gf , . This, in fact represents the interaction of f and g . And the interaction can be different if we change the order. Definition 2.4. The measure of interaction of a pair Μ ∈ gf , will be denoted by ),( gfp and defined as )1()(),( ggfgfp γγ =. If γ is defined as in Theorem 2.1 we get  Ω = µττ dgfgfp )()(),( 21 where 1 τ and 2 τ are given transformations Τ such that )(1)( 22 gg ττ −= ∗ . Theorem 2.3. Let p be a measure of interaction on the MV-algebra Μ and gf , be any elements of Μ . Then )(),1()1,( ffpfp ν = = 0),(),( = = fgpgfp , if f and g are ν -orthogonal ))(),(min(),( gfgfp ν ν ≤ , particularly )(),( fffp ν ≤ the variables of p do not commute, i.e. in general ),(),( fgpgfp ≠ . Example 2.1. Assume that ]1,0[ = Ω and µ is the Lebesgue measure. For any element Μ ∈ f put ff =)( 2 τ and the transformation 1 τ will be defined by the following          <<= = = =  }.1)(0;{ , )( )( 0)(0 1)(1 ))(( 1 xfxA whereotherwise A dxf xf xf xf A µ µ τ Let ( ) xxf = and )5.0,0max()( − = xxg . Then 0 = x 1 = x 10 < < x )))((( 1 xf τ 0 1 5.0 5.0 ≤ x 5.0 > x )))((( 1 xg τ 0 25.0 Then  =−== 1 5.0 1 0 16 1 )5.0(5.05.0),( µµ dxgdgfp . 32 3 25.0),( 1 5.0 ==  µ xdfgp 16 3 5.0),( 1 0 ==  µ xdffp 16 1 )5.0(25.0),( 1 5.0 =−=  µ dxggp 3. SOME COMMENTS CONCERNING OBSERVABLES AND THEIR JOINT DISTRIBUTION First we recall the definition of a tribe and of an observable. Definition 3.1. An MV-algebra Μ will be called a tribe iff for any non-decreasing sequence of elements ∞ =1 }{ i i f there holds Μ∈= ∨ ∞ = ff i i1 . From now on we will assume the MV-algebra to be a tribe. Definition 3.2. An observable is a mappping λ from Borel sets )(R Β into the MV-algebra Μ such that .1)( = R λ If 0 / = ∩ BA , then )()()( BABA λ λ λ ⊕ = ∪ and ).()( ∗ ≤BA λλ If n A  A , then )( n A λ )(A λ . Conditional measures on MV-algebras 33 In a natural way for each observable λ we can define also its cummulative distributive function λ F and its expectation )( λ E by ])),((()( xxF −∞= λν λ )()( dxFxE  ∞ ∞− = λ λ . And, making a parallel to the measure of interaction p from Definition 2.4, we can define the joint probability distribution λκ P for any pair of observables λ and κ by ))(),((),( BApBAP κλ λκ = where BA, are Borel sets. This can be interpreted as the measure of interaction of the observables λ and κ . The basic properties of λκ P can be just rewritten from Theorem 2.2. It is also possible to define the mean interaction of the observables λ and κ , denoted by ),( κ λ C, as follows  ∞ ∞− −−= ),())())(((),( , xxdFExExC κλ κλκλ where ]),(],,((),( ,, yxPyxF −∞−∞= κλκλ is the joint cummulative probability distribution. The investigation of the joint probability distributions (the interactions) of observables and of the corresponding mean interactions will be the topic of a next paper. Here we would like to point just to one very important property of the introduced notions, namely to the non-commutativity of the variables (observables) in the measure of interaction κλ, F and in the mean interaction ),( κ λ C. REFERENCES [1] C.C. CHANG: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467490 [2] C.C. CHANG: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74-80. [3] F. Chovanec: States and observables on MValgebras. Tatra Mountains Math. Publ. 3 (1993), 55 -63. [4] M. Jureková, B. Riean, Weak law of large numbers for weak observables in MV algebras, Tatra Mountains Math. Publ. 12 (1997), 221 -- 228. [5] O. Nánásiová: A note to independent events on quantum logic. Busefal, 76, (1998), 53-57. [6] O. Nánásiová: Principle conditioning. Sent to Int. Jour. of Theor. Phys. (2000) [7] O. Nánásiová, A. Yu. Khrennikov: Observables on a quantum logic. Foundations of Probability an Physics -2, Conf. Proc. 005, ICMM, Wajxo, Sweeden, (2002), 417-431. [8] O. Nánásiová: Map for simultaneusous measurements for a quantum logic. Int. Jour. Of Theor. Phys. 42, (2003), 1998-1902. [9] B. Riean: On the sum of observables in MV algebras of fuzzy sets. Tatra Mountains Math. Publ. 14, (1998), 225232. [10] B. Riean: On the strong law of large numbers for weak observables in MV algebras. Tatra Mountains Math. Publ. 15 (1998), 1321. [11] B. Riean: Weak observables in MV algebras. Internat. J Theoret. Phys. 37 (1998), 183 - 189. [12] B. Riean: On the product MV algebras. Tatra Mountains Math. Publ. 16 (1999), 143 - 149. [13] B. Riean, T. Neubrun: Integral, Measure, and Ordering. Kluwer Academic Publishers, Ister Science, Bratislava, (1997).