The Jegorov theorem on MV algebras

5 y ~ z 7 .~- - er,r.

FUII'Y

sets and systems ELSEVIER

Fuzzy Sets and Systems 105 (1999) 307-309

The Jegorov theorem on MV algebras M f i r i a Jure~kov~i Military Academy, Department of Mathematics, SK-03119 Liptovskf Mikul~L Slovak Republic Received July 1998

Abstract The notion of the ®-compatible sequence of observables on MV algebras is given in this paper, and a version of the Jegorov theorem for this sequence of observables is presented. @ 1999 Elsevier Science B.V. All rights reserved.

Keywords: MV algebra; Jegorov theorem

1. Introduction First let us consider an algebraic system ( J / , @, G, *, 0.a, 1.a), where ,//¢ is a nonempty set, ~3, ® are the binary operations, • is the unary operation and 0¢/, Ira are fixed elements o f .//¢ such that the following axioms are satisfied: a ~ b = b ~ a, a@(b@c)=(a@b)@c, a@O=a, a~31ug=lt¢, (a*)* = a, 0"/¢= 1m, a ~3 a* = 1°~, (a* @ b)* @ b =

(a@b*)* @a, a@b=(a* @b*)*. This algebraic system J / is called an M V alge-

bra [1]. Example 1. Every Boolean algebra is an MV algebra. Especially, if ~ is an algebra o f subset o f a nonempty set X and we put E @ F = E U F , E @ F = E N F , E* = X \ E , for every E, F E ~ , 0 = 0, 1 = X , then is an MV algebra. Example 2. I f we consider a set ~ c (0, 1)x, closed under the binary operations f ~ 9 = m i n ( f + g , 1), f @ g = m a x ( f + 9 - 1, 0), where ( f + 9)(t) = f ( t ) +

g(t), for every t E X and the unary operation f * = 1 - f , containing the constant function 1x, then we obtain a typical example o f an MV algebra. If we define a V b = ( a G b * ) ® b , a A b = ( a ® b * ) G b and a ~
0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PIE S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 3 3 0 - 3

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M. Jure(kovdt/Fuzzy Sets and Systems 105 (1999) 307-309

2. The Jegorov theorem Definition 1. Let ~ be an MV a-algebra. A sequence (x,), of observables xn : ~ ( R ) ~ ~ ' is called 6)-compatible, if for every J c N , J = { i ~ .... ,ik} there exists a mapping w j : ~ ( R Idl )--. ~/¢ such that the following properties are satisfied: (1.1) wj(glJI)= 1~; (1.2) if -4,An~ ~(RIJI), -4n/~-4, then wj(-4,) /

Wj(A); (1.3) if,'/E g~(RIJI ), then WJ( ~ IJ I \-4) ~- WJ( -4 )*'~ ( 1.4) if,,/, B ~ ~ ( R IJ I), -4 ~ B = 0, then wj (-4 U B ) = WJ(-4 ) @ w J ( B )'~ (l.5) Wd(-4il × "'" × Aik ) -~- Xil(-4il ) 6) "'" 6)

xi~(A~,), for all Ai, . . . . . -4i, 6 ~#(R). For example, any sequence of observables from Example 1 is 6)-compatible.

Let us put .£e = {-4 C ~(RIJ, I); w j 2 ( ~ 1, (-4)) = wj, (-4)} and denote by 9 the family of all rectangles -4t~ × "'" × -4tk; -4tl . . . . . -4tk E ~ ( R ). Evidently .L# ~ 9 . From the properties of the mapping wj it follows that .W is a q - a-algebra over the ring ~ ( 9 ) generated by 9 . Therefore, .W ~ q - a(~ ( 9 ) ) = a(~ ( 9 ) ) = . ~ ( R ]J'l ) which implies wj~(~,

(-4) ) = wj,(-4)

whenever A E ~(RIJ, I).

Fix a state m on J / . According to property (1.8) of the mapping w j, we are able to define a consistency system of probability measures {Pj, 0 # J C N} defined via Pj(-4)=m(wj(.4)),

Lemma 2. A mappino W j ( ~ ( R IJ[ )) ~ ~ has the followin9 properties: (1.6) ijr - 4 , B E ~ ( R I J [ ) , - 4 C B , then w j ( B \ A ) = w j ( B ) ~ wj (-4); (1.7) if - 4 E ~ ( R ) , then W j ( { ( t l . . . . . ti . . . . . tk)E ~ ( R IJI ), ti ~ A} ) : xi(A); (1.8) if J~ C J2 c N , then wj2(~z~lt(-4))=wj~(-4), for all-4 e ~(RId~ [), where nj2j~ : ~(gId21 ) --, ~(RIJ, I), is the projection.

Proof. (1.6) is proved in [4]. (1.7) WJ({(tl . . . . . ti . . . . . tk) E ~(R IJI), ti E-4}) = w j ( R × ... × R x - 4 × R . . . × R ) = x t ( R ) ® . . . 6 ) Xi-l(R)6)xi(-4)6)xi+l(R)6) " " 6)xk(R) = 1 ~ 6 ) . . . 6) 1.~t ® xi(A) 6) 1~ 6) ... 6) l~a = xi(A). (1.8) Let Jl Q J z Q N ,

-4=-4t~ x . . . ×-4tk E~(RIJI[),

J2JI(-4)-~-R × "'' x R ×At, x ... ×-4t~ x R

× ... ×REgf(RtJ21),

wj~ ( ~ l, (-4)) = x,, (R) 6)... 6)x~,(R) 6)x,, (-4,,)

= 1~ 6 ) . . . 6) l~a 6)xt~(-4tm) 6 ) ' " 6) xtk(-4t~) 6) 1~ ® . . . 6) 1~ = w j, (-4).

A E ~(RIJ'I).

It is not difficult to prove that Pj is a probability measure and it holds:

Pj, (-4) = PJ2(~l, (-4)), where 7~J2Jt : R IJ21--* R JJ~I is the projection, Jl C Jz C N, .4 E ~(RIJll ). Let R N be the set of all sequences of real numbers, ~zj:RN---*R IJ[ be the projection, i.e. ~zj((tn)n)=(tj~ . . . . . tjk ) for J : j l . . . . . jk. By the Kolmogorov theorem there exists exactly one probability measure P on the measurable space (R N, a(v)) (v is the family of all cylinders, i.e. the set of all sets o f the form ~ f l ( A ) , J c N , A E ~(RIJI), and a(v) is the a-algebra over v) satisfying the equality

P ( ~ j t ( A ) ) =Pj(A) for all d E ,.~(R [J[ ) and every finite J C N. Let us define the mapping ¢i : RN --+ R; ~i( ( tn )n ) = ti, i = 1,2 . . . . . Evidently ~i is a random variable, and it holds:

P~,(-4) =

6) . . . 6) x,A -4 ,~ ) 6) x~, ( R ) 6 ) . . . 6)x,.(R )

[]

=

e(~i--l(-4))

~-~ P({(tn)n;

ti E-4)

P(~.{(-4)) = m(w{i}(-4))

= m ( x i ( A ) ) = taxi(-4).

According to the preceding procedure, we can construct a sequence (~n)n of the random variables for any sequence (xn), of observables.

M. JurePkov(t/ Fuzzy Sets and Systems 105 (1999) 307-309 Now, we can start with the Jegorov theorem. Let us mention one of the classical version of this theorem. The Jegorov theorem asserts [4]: If a sequence of the random variables defined on a probability space converges almost everywhere to 0, then it converges to 0 almost uniformly. It is necessary to introduce the above-mentioned types of the convergences for the sequence of the observables on MV algebras. The motivation for the definition of the almost everywhere convergence of the sequence of the observables on MV a-algebras is the almost everywhere convergence of the sequence of the random variables on a probability space. Definition 3. A sequence (x,), of observables on an MV a-algebra J / c o n v e r g e s m-almost everywhere to the zero observable 0, if

(( 1,p Z~I ,in, Hm imm,~n~=k{k+i x p~ock~i---*~ k\ p/ ) - - , We note, that the zero observable 0 is the observable such that O(E)=

Proposition 4. Let (x,), be Q-compatible sequence of the observables on an MV a-algebra J/I, (~,), the corresponding sequence of the projections defined on (RS,a(v),P). (Xn), converges m-almost everywhere to 0 if and only if (in), converges P-almost everywhere to O. Proof. Let (t,), = ?, and ~i :RN---->R be a random variable such that: ~i(t-'): ti for all i E N. Then

1})

t; I~.(K)l < P

=m

Vb > 0 3c E JP[; re(c) > 1 -- b, Ve>O 3 k E N ; Vn>~k Ec., m(c.)/c.+l >~c,

c @ c. <<.x.((-e, ~)). This definition was used in [3]. Rie~an in this paper proved that if (x,), is a sequence of observables on the D a-poset ~ (an MV a-algebra is one of the examples of D a-poset) such that for every finite set J C N there exists the mapping wj : ~ ( R IJI ) ---+~ satisfying the above-mentioned properties (1.1), (1.2), (1.6), ( 1.8) then the almost everywhere convergence of the sequence (¢,), to 0 implies the almost uniformly convergence (X,),eN to the zero observable 0. By using this assertion and the above-mentioned Proposition 4, we can prove the Jegorov theorem for a sequence of Q-compatible observables on an MV a-algebra.

(tk . . . . . tk+i); - - < Pt , < -

Proof. Let (x,), be a sequence of ®-compatible observables which converges m-almost everywhere to the zero observable 0 and (~,)n be the sequence of corresponding projections on (R N, a(v), P). According to Proposition 5, (3,), converges P-almost everywhere to 0. Now, we use the classical Jegorov theorem for the random variables and above-mentioned assertion from [3] and we can make the conclusion that (X,),eN converges to 0 m-almost uniformly. []

References

\ n:k

=Pj

Definition 5. A sequence (x,), of observables on an MV a-algebra J ¢ converges m-almost uniformly to the zero observable 0 if:

Theorem 6. Let JJ be an M V a-algebra and (x,), be a sequence of Q-compatible observables on J[. I f (x,), converges m-almost everywhere to O, then it converges to 0 m-almost uniformly.

1, 0 E E , 0, O ~ E

for any E E ~(R).

P

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P

[l] C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958) 467-490. [2] F. Chovanec, States and observables on MV algebras, Tatra Mountains Math. Publ. 3 (1993) 55-63. [3] B. Rie6an, On the convergence of observables in D-posets, Tatra Mountains Math. Publ. 12 (1997) 7--12. [4] B. Rie~an, T. Neubrunn, Integral, Measure, and Ordering, Kluwer Academic Publishers, Dordrecht, Ister Science, Bratislava, 1997.