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In a text I read the following:

"Let P be a translation invariant product measure on $\left\{0,1,2\right\}^{\mathbb{Z}^d}$ in which each state has positive probability."

(1)

I know that for the Borel$-\sigma$-Algebra $\mathcal{B}$ on $\left\{0,1,2\right\}^{\mathbb{Z}^d}$, which can be generated by the cylinder sets $$ \text{cyl}(y_{i}^{i+n})=\left\{x\in\left\{0,1,2\right\}^{\mathbb{Z}^d}: x_i=y_i,x_{i+1}=y_{i+1},\ldots,x_{i+n}=y_{i+n}\right\}, $$

one can define product measure P in which each state has positive probability, which I call $p(0),p(1).p(2)>0$, $p(0)+p(1)+p(2)=1$ as follows $$ P(\text{cyl}(y_{i}^{i+n}))=\prod_{k=i}^{i+n}p(y_k), $$ Then this kind of product measures is always translation-invariant, right? Because for the cylinder sets that is fullfilled.

Are there more such measures?

(2) But I do not know a non translation-invariant product measure on $\left\{0,1,2\right\}^{\mathbb{Z}^d}$ in which each state has positive probability.

Are there such measures? And on which $\sigma$-Algebra?

  • If $x : \mathbb{Z}^d \rightarrow {0,1,2}$, then what is $x_i$? – snar Jan 02 '15 at 18:16
  • Oh sorry. The index should be d-dimensional. –  Jan 02 '15 at 18:17
  • Still not sure what you mean. Presumably $y_i$ is a scalar in ${0,1,2}$. If you want to define a probability space $\Omega$ and random variables $X_i : \Omega \rightarrow {0,1,2}$ such that for all $i, n \in \mathbb{Z}$, $P(X_i = y_i, \dots, X_{i+n} = y_{i+n}) = \prod_{k=i}^{i+n} p(y_k)$, then $\Omega = {0,1,2}^\mathbb{Z}$ will suffice. – snar Jan 02 '15 at 18:25
  • The cylindersets are 0,1,2 points on the larice where finite many are determined by the y_i values which are 0,1,2. –  Jan 02 '15 at 18:29
  • What do you call $p(0)$ when $P$ is not shift-invariant? – Did Jan 02 '15 at 18:31
  • Hm, do not see what you mean. With p(0) I mean the probability for 0 which is used in the product measure. –  Jan 02 '15 at 18:33
  • I just ask that because in a text I read "let P be a translation invariant product measure on $\left{0,1,2\right}^{\mathbb{Z}^d}$ in which each state has positive probability." So I wondered if there are such measures that are NOT translation invariant. Maybe it is meant on other sigma algebras than the one generated by the cylindersets? –  Jan 02 '15 at 18:47
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    What do you call p(0) when P is not shift-invariant? Note that "the probability for 0 which is used in the product measure" is absurd since the product measure is a measure on a set which does not contain 0. – Did Jan 03 '15 at 01:21

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