I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a standard course in (possibly multivariate) statistical inference.
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I have just the right book for you. Try Theory of Statistics by Michael J. Schervish. It is the only book on measure-theoretic statistics that has received $ 5 $ stars from every person who commented on it on Amazon.
I also suggest reading the Lehmann volumes, Theory of Point Estimation and Testing Statistical Hypotheses, which do not compromise on mathematical rigor although Erich Lehmann was a statistician.
Professor Dudley of MIT has a wonderful set of notes on measure-theoretic statistics, which I personally refer to very often. If you would like to see more, then Dennis Cox of Rice University has a set of notes entitled The Theory of Statistics and Its Applications.
Finally, let me say that it is important to read landmark papers that have introduced some of the major concepts that we see in the theory of statistics today. Having said this, I highly recommend Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics by Paul Halmos and Leonard Savage.