There are two questions about measures bothered me a lot.

- Given a set X and a countable covering ${U_i}$ of $X$. Suppose that for each i, there is a measure $m_i$ on $U_i$. Is there a very general procedure to define a non-trivial measure $m$ on X by using all of $m_i$´s?

Remark: Each $(U_i,m_i)$ induce a measure $Ind_i$ on $X$ by the obvious way, so by non-trivial, I mean the measures which are not produced from this way. If this covering is finite, then of course we can define a measure via addition of measures. If the intersection $U_{ij} = U_i \cap U_j$ satisfied that $m_k(U_{ij}) = 0$ for $k = i,j$, then we can also use addition of measure. But for infinite covering such that the intersection may have positive measures, I have no ideas.

- Given a group $G$ acts on a measure space $(X,m)$ with $m$ is a $G$-invariant non-atomic measure. Is it possible to define a measure on the quotient space $G/X$?

Remark: If $G$ acts on a manifold $X$ equipped with a $G-$invariant measure properly discontinuous, then this can be done.

somemeasure, and $\infty$ seems as well justified to me as any other. $\endgroup$ – LSpice Mar 9 at 18:12