Consider $\mathbb{Z}^d$ as a graph $(\mathbb{Z}^d, E(\mathbb{Z}^d))$ such that there is an edge between $x,y$ if and only if $\|x-y\|_1=1$
So I am currently working with a measure on $\mathbb{N}^{E(\mathbb{Z}^d)}$ that is the limit of measures in $\mathbb{N}^{E(\Lambda)}$ where $\Lambda \subset \mathbb{Z}^d $ is finite subgraph. I managed to show that the $\mathbb{P}_{\Lambda}(A)$ has a limit for every $A$ event that depend on a finite number of edges.
To show that there is this defines a measure $\mathbb{P}$ in the whole $\mathbb{N}^{E(\mathbb{Z}^d)}$ I need some kind of extension theorem. My first though was to use one of those:
- Caratheodory Extension Theorem: The problem is to justify the $\sigma$- aditivity of the measure in the algebra of the events that only depend on a finite number of edges. Exchanging the order of limit and the sum in $ \mathbb{P}(\cup_n A_n) = \lim_{\Lambda} \sum_{n} \mathbb{P}_\Lambda(A_n) = \sum_{n} \mathbb{P}_\Lambda(A_n) = \sum_{n} \mathbb{P}(A_n)$
- Monotone Class Theorem: Proving that would work for monotone sequences of events doesn't seen any easier
- Riesz-Markov-Kakutani Theorem: That seemed to be my best shot, as I could construct the functional that sends the local functions to their mean value, extend this functional by density and finally use the representation theorem. But the theorem request that the space is locally compact, which is not the case.
Would anyone have another suggestion of a approach to extend such a measure?