Probability Examples and Explanations

Recent questions in Probability
High school probabilityAnswered question
Ghillardi4Pi Ghillardi4Pi 2022-11-26

What does a cross product of 0 mean?

High school probabilityAnswered question
Siemensueqw Siemensueqw 2022-11-13

Measurable and non-singular equivalence relation in a one-sided Markov Shift
In a one-sided Markov shift ( X , B , μ , T ) where X = { 1 , 2 , , n } N is the sample space, B the σ-algebra generated by finite cylinder sets, μ a product measure and T the shift transformation, first, we define a metric d ( x , y ) = 2 n where n is the least integer such that x [ n ] y [ n ]. Then, we define the following relation:
Given x , y X, x r y iff there exists n , m Z 0 such that T n x = T m y.
One can easily check r is an equivalent relation. We set R = { ( x , y ) X × X | x r y } and, for each subset A X, we set R ( A ) = a A [ a ] r to be the union of equivalent classes of each element in A.
My question is:
1. Is R B × B = { A × B | A , B B }?
2. For each B B , is it true that μ ( B ) = 0 iff μ [ R ( B ) ] = 0?
The first question is asking if R is measurable and the second is asking if R is non-singular. For the first one, it suffices to show that, for each B B , R ( B ) B . However, R(B) could be properly contained in n N T n ( B ) and then I do not know how to proceed. I also try to consider the function ( x , y ) inf n , m d ( T n x , T m y ). This function is continuous but it may obtain zero outside of R.
For the second question, I suppose whenever μ ( B ) > 0, B will contain a finite cylinder set and hence so does R(B). However, I do not how to deal with the other direction.

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