If you search economics book there are plenty of them on the market. We may say the say about "mathematics for economists" or "application of mathematics in economics". But I've never seen book which tells me about economics based on statistics, optimization and other field of mathematics. Where every law, curve and so on is based on some statistical experiment or real analysis.
Example: I'm currently reading https://www.amazon.com/Principles-Microeconomics-N-Gregory-Mankiw/dp/1305971493 and the problem here is that there're no proof for principles. Author just says "people always have trade-offs". But from the other hand there's optimization theory, and if we care only about revenue function - we may find maximum. (So in that case we don't have any trade-offs).
Let me know, please, if you know any book "economics for mathematicians". Thanks!
PS. There's similar question on site, but it's outdated. (But have a little different idea) (Textbooks of economics for mathematicians)
Efe Ok, Real Analysis with Economic Applications provides an accessible introduction to aspects of real analysis that are important for economists. It's particularly strong on things to do with expected utility theory and optimization. It's quite a bit easier than standard graduate school texts covering real analysis. He also has a parallel work in progress on probability, available from his website.