Suppose $E \rightarrow M$ is a vector bundle, and $G$ a Lie group acting by vector bundle automorphisms on $E$, such that the action on the base is proper. Is it true that given any smooth section $\sigma$ of $E$ one can construct a $G$-invariant section of $E$?
I found something in a paper where a special section was needed to give you some nice construction. This section was proved to exist, and then there was the comment that if you have a $G$-action, one can choose a $G$-invariant open cover of $M$, and make the section $\sigma$ $G$-invariant over each open set of the cover by averaging, using that the stabilizer groups are compact. Finally, you glue everything together with a $G$-invariant partition of unity. This would give a $G$-invariant version of the result.
I am not familiar with this "averaging" process. Could you explain what is really meant?