Ergodic theorem for ergodic mapping on B-structures
by
Lenka Lasova
Matej Bel University Banska Bystrica
Coauthors: Magdalena Rencova (Matej Bel University Banska Bystrica)

In this contribution we will show the extended ergodic theorem for B-structures with a state. Classic ergodic theorem is determined for ergodic mapping on W, where (W, S, P) is a probability space and x:W→ R is an integrable random variable. In our case W is replaced by a B-structure B, which is defined as a system (B, [^(⊕)], ≤ , 0_B, 1_B) such that:

(i) [^(⊕)] is a partial binary operation on B;

(ii) ≤ is a partial ordering on B;

(iii) 0_B is the smallest, 1_B is the largest element in (B, ≤ ).

By a state on B we mean a mapping m:B→[0, 1] satisfying the following conditions:

(I) m(1_B)=1, m(0_B)=0

(II) if a=b[^(⊕)] c, then m(a)=m(b)+m(c)

(III) if a_n\nearrow a, then m(a_n)\nearrow m(a).

In the next text we denoted by B(R) the family of all Borel sets.

Instead of integrable random variables x:W→ R we use integrable observables on B. For observable we will take a mapping x:B(R)→ B, which satisfy following:

(i) x(R)=1_B, x(∅)=0_B;

(ii) A, B ∈ B(R) and A∩B=∅ then x(A∪B)=x(A)[^(⊕)]x(B);

(iii) if A_n ∈ B(R):A_n\nearrow A then x(A_n)\nearrow x(A).

It is integrable if exists ∫_Rtdm_x(t), where m_x=m○x with m as a state on B.

So the ergodic theorem is:

Let x be integrable observable on B-structure B with state m, for which the following holds: ∀a ∈ B: m(l(a))=m(a) and l:B→ B is ergodic mapping. Then the sequence (y_n)^∞_n=1 defined by a formula:

y_n=[1/n]∑^n-1_i=0lⁱ○x - E(x)

converges m-almost everywhere to 0.

References:

[1] K. Cunderl\' ikov\' a-Lendelov\' a, B. Riecan: Probability on B-structures. In Fuzzy sets and systems accepted, 2007.

[2] T. Neubrunn, B. Riecan: Integral, measure and ordering. Dordrecht, Kluwer 1997.

[3] B. Riecan: Representation of probabilities on IFS events. Advances in Soft Computing, Soft Methodology and Random Information Systems. Springer, Berlin 2004, 234-246.

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Date received: January 30, 2008