Variational principle for GR

My earlier question about up-dated accounts of Emmy Noether's own
version(s) of the Noether conservation theorem(s) gets a thorough
answer (with appreciative historical remarks on Lie and Noether) in
Peter Olver's book Applications of Lie Groups to Differential
Equations. But I am still struggling with the relation to GR.

As I understand it, one approach to GR is to define a Lagrangian
function of the metric tensor, called the action of the metric. Then
the Einstein field equation is derived by finding a metric field that
minimizes the action. Is that right, so far as it goes? Anyway, it
is vague.

Where is the integral taken? Some formulations seem to say that this
only works for aymptotically flat regions in space time, and then you
integrate over any region which is flat all around the boundary.

I suppose that in a compact space-time you could integrate over all of
space-time. Or have I completely stopped making sense when I say
that?

I would appreciate any answers or references. I have not mastered GR
but I have read the first 4 chapters of Wald, General Relativity, and
have read Misner Wheeler and Thorne through "How mass-energy generates
curvature (chap 17)." I hope I need not actually read either one of
those books all the way up to where they do Lagrangian formulations.

Colin

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