Measure on union of measure spaces and on quotient space 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.  
 A: The first question seems too broad because there are a lot of things you can do. If they are probability measures you can take a weighted sum such as $\sum \frac{1}{2^n}m_n$. Or you could disjointify by letting $V_n = U_n\setminus(U_1 \cup \cdots \cup U_{n-1})$, restricting each $m_n$ to $V_n$, and summing up. It really depends on what you need this for.
For the second question, a nice way to look at this is by considering the dual action of $G$ on $L^\infty(X)$. If it is finite (or a probability measure), the $G$-invariant measure on $X$ translates to a normal $G$-invariant linear functional on $L^\infty(X)$ (or a normal state), and passing to the quotient corresponds to passing to the subalgebra of $G$-invariant functions in $L^\infty(X)$. This is a von Neumann subalgebra of $L^\infty(X)$, so it is isomorphic to some $L^\infty(Y)$, and you can just restrict the original linear functional on $L^\infty(X)$ to the subalgebra to get a normal linear functional on $L^\infty(Y)$, i.e., a measure on $Y$.
Morally $Y$ is the quotient of $X$ by the $G$-action, but I'm not sure whether you can make this literally true. It seems likely that some kind of measure theoretic pathology would block this from working in general.
