I have read a bunch of book on Game Theory, and I find that one of the best is the the book of Osborne and Rubinstein. Nevertheless, all books which I found are not made for mathematician, meaning that they do not assume prior knowledge, in particular they do not assume measure theory, functional/complex analysis etc.
Is there any really formal and well-written book about game theory made for mathematician? Thanks in advance
I do not know what you mean by prior knowledge, but I would give a couple of books which requires more mathematical sophistication than Osborne and Rubinstein. If you are looking for a specific area in game theory then this list may not be useful for you.
Game Theory by Maschler, Solan and Zamir: This book is relatively new and it includes almost everything one needs to know about game theory (both cooperative and non-cooperative). In the mathematical appendix it covers necessary background for existence theorem.
Game Theory for Economic Analysis by Ichiishi: This is a pretty old book, but it is very well written and it assumes a great deal of mathematical knowledge. Yet, the initial chapters discuss continuity notions for set valued maps(uhc-lhc), properties of convex sets and fixed points.
Game Theory by Owen: This is also another old book, but it is still used in various courses in different places. It has traditional definition-theorem-proof structure.