Is there a mathematically rigorous book giving an introduction to boosting, etc. A book that is rigorous like "A Course in Probability Theory" by Kai Lai Chung.
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Along with the text that @William proposed (it's a great reference), for bootstrapping it's hard to beat:
Efron and Tibshirani 1994, An Introduction to the Bootstrap.
Hall 1995, The Bootstrap and Edgeworth Expansion is more rigorous than the above. Upon a cursory glance, it appears to dive pretty deep into the theoretical details of the bootstrap.
Specifically for boosting, Robert Schapire has a list of references to read found here. And for bagging, Martin Sewell has a list of references here. His site has a bunch of reference lists for a number of machine learning topics (see here).
Elements of Statistical Learning is available online.
There is a chapter on Boosting (chapter 10).
Does it satisfies your definition of rigorous? If by "rigorous" you mean "measure theoretic," then no.
Shapire's Boosting: Foundations and Algorithms is IMHO a very didatic and rigorous book about the subject.