Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions concerning the following situation:
For any natural number $k$, and any elements $\{ x_1 \in X_1, …, x_k \in X_k \}$, existence of the relation for any two of the elements and existence of the relation for these $k$ elements imply each other? In other words, existence of pairwise relation and existence of mutual relation are equivalent?
For example,
In probability theory, for a (finite, countably infinite, uncountably infinite) set of events, mutual independence implies pairwise independence, but pairwise independence does not imply mutual independence. I was wondering why? Specifically what kind of property does measure space lack to make the two equivalent?
Thanks and regards!