From my own experience, I will advise you against every book of mathematical methods written specifically for physicist. From my point of view, is better to learn about mathematics from mathematically written books (it sounds so obvious but is not). For example, many people like Schultz, Geometrical methods of mathematical physics, but I prefer to learn about the common topics in Singer, Thorpe, Lecture notes on elementary topology and geometry. (I don't say it is not a good textbook, I only say I find difficult learning things on books written in a pretty informal way.)
The most complete work about methods of mathematical physics is probably
- Reed, Simon, Methods of modern mathematical physics,
that covers functional analysis, Fourier analysis, scattering theory, operator theory.
Since you are interested in cosmology, the best review on Loop Quantum Gravity is that by Thomas Thiemann,
- Thomas Thiemann, Modern and canonical quantum general relativity,
a 900 pages review, equipped with about 300 pages of mathematical methods (mathematical appendices are not a textbook however, but a collection of necessary results, eventually explored in some depth). References therein are very useful also.
Many people like
- Deligne et al., Quantum fields and strings: a course for mathematicians,
that joins a good part of your requests. (I haven't read it, however, I know it since is "famous".)
A celebrated book on methods of classical mechanics, concerning manifolds too, is
- Abraham, Mardsen, Foundations of mechanics.
Another is
- Choquet, Bruhat, Analysis, manifolds and physics.
All Arnold's books are always a great choice. (he wrote about ergodic theory and geometrical methods for differential equations, among the other things.)
There are a lots of more specific books, e.g. dealing with mathematical structure of quantum mechanics, but many of those are more and more specialized and is better to have very clear the general theory before try to get more involved into dangerous subjects such as, to say, quantum field theory. Once one has a strong background, the best opera on the subject of field theory probably is
- Zeidler, Quantum field theory,
an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,
- Haag, Local quantum physics.
EDIT. I'd like to add some book I've discovered more recently and I think fit very well:
Streater, Wightman, "PCT, Spin and all that",
Teschl, "Mathematical methods in Quantum Mechanics",
Bogolioubov, Logunov, Todorov, "Axiomatic Quantum Field Theory",
Lansdman, "Mathematical concepts between classical and quantum mechanics".
Best Answer
There are two versions of such path integrals: the Minkowskian and the Euclidean. The first one is of the form $$ \int\ F(\phi)\ e^{iS(\phi)}\ D\phi $$ and the second one is $$ \int F(\phi)\ e^{-S(\phi)}\ D\phi\ . $$ Here $S$ is the action functional and $F$ is another functional corresponding to observables. The Minkowskian case lies outside ordinary measure theory, even in the Gaussian case (a Theorem by Cameron https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1960391126 correcting an earlier mistake by Gelfand and Yaglom who thought that $\sigma$-additive complex measures would work). One instead has to use a limit of a time slicing procedure to make sense of it.
For the Euclidean case, ordinary measure theory is perfectly adequate. The "I heard that as traditional measures it is not possible" is a common misconception that unfortunately gets repeated ad infinitum. The part $e^{-S(\phi)}\ D\phi$ should be a Borel probability measure on a space of distributions like $\mathscr{S}'(\mathbb{R}^d)$ seen as an ordinary topological space. The most canonical topology to use is the strong topology. When dealing with concrete models, it is a highly nontrivial task to construct this probability measure which a priori should be the weak limit of a sequence of well defined probability measures obtained by introducing ultraviolet and infrared cut-offs.
Edit: For more details on how to construct the measure as weak limits of (finite) lattice measures in the free field case see:
https://mathoverflow.net/questions/362040/reformulation-construction-of-thermodynamic-limit-for-gff
https://mathoverflow.net/questions/364470/a-set-of-questions-on-continuous-gaussian-free-fields-gff?noredirect=1&lq=1
https://mathoverflow.net/questions/384124/bochner-minlos-for-moment-generating-functions?noredirect=1&lq=1