These don't have anything on spectral analysis, but cover the rest pretty well: Infinite Dimensional Analysis: A Hitchhiker's Guide by Aliprantis and Border, and Stocastic Stochastic Limit Theory by Davidson. The first book was written for economic theorists, the second one for econometricians. So they specialize quite nicely to what you seem to be working on. However, they are both rather advanced.
A great introduction to measure theoretic probability is Probability with Martingales by Williams. It's quite chatty and fun, but does still require some mathematical sophistication. A book that I think is a bit dry but that proceeds in small and easy steps with all the details included is A Probability Path by Resnick.
