Is it possible to build a larger sigma algebra than caratheodory process will yield ( not in euclidean spaces) but in more general abstract setting?
I don't know the answer, but Heinz König has replaced the Carathéodory construction by a different one which seems to be superior. You can have a feel of König's ideas by looking at the attached paper, or other papers of his.
Article The New Maximal Measures for Stochastic Processes
If I understand correctly, what you are looking for is a method to generate measures that yields measures that are defined on a larger sigma-algebra compared to the one obtained using the Carathéodory construction. I am not sure this is possible in general, and it may really depend on what you are starting from. There are several ways of generating measures. First you can construct an outer (or an inner measure). Then using these you can construct a measure. You can construct outer measures using the covering approach (the standard Caratheodory one,) and the packing approach. Maybe the packing approach may be useful to you. Here you can find something about it : http://classicalrealanalysis.info/documents/BBT-AlllChapters-Landscape.pdf
ine can have a larger sigma algebra than carathedory construction by using gauge integration.
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