Measure and category
by John C. Oxtoby, Oxtoby · first published 1971
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…case, there is Moore-Smith convergence, nets, filters, etc. Somewhere in there he will discover the continuum hypothesis, model theory, etc. But under no circumstances let him get near John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces', ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971. If he sees this book, he may never recover!
on “Indescribable numbers: The theorem that made me fall in love with math” ↗…course in analysis and probability. So, the analysis part was, right, basically Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. There was also some material from Oxtoby, Measure and Category (ice cream and cake dessert -- super fun stuff). The probability was right from the beginning sigma algebras, etc. So, a central topic was the Radon-Nikodym theorem and conditional expectation -- gorgeous…
on “Ask HN: What are your favorite statistics and probability textbooks?” ↗…exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory,…
on “Ask HN: Older textbooks/papers you consider classics still worth studying today?” ↗Measure and Category by John Oxtoby. This book studies duality results between different notions of "small" sets in measure theory and topology. It's the first (and to some extent the only) math book where things just clicked…
on “Ask HN: Math books that made you significantly better at math?” ↗…e.g., in parts of engineering, is not trivial and is made easy by Luenberger. It's a lesson: Some of pure math can be much more powerful in applications than is easy to see at first. John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces' Elegant. Astounding. Some of what learn via the Baire category theorem can shake one's intuitive view of the real line and our 3-space.…
on “What's a Mathematician to do?” ↗…more general. Then notice the Carathéodory extension result -- that is the key to defining measures on the real line, in particular, Lebesgue measure. Spend at least a weekend in John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \ An amazing weekend. Keep at hand Bernard R.\ Gelbaum and John M.\ H.\ Olmsted,…
on “Ask HN: Resources to learn real analysis?” ↗What's your take on this? Got a counter proof to Cantor? As much as I am a fan of all things about cardinals, measure and category (Oxtoby's booklet on this stuff has been on my nightstand for several months, to a great pleasure), I still fail to recognize its technological and physical implications. My take on this is modelled…
on “Why Discrete Math Is Important” ↗Yes, the set of reals, as in the usual construction, is the only complete Archimedean ordered field. And some amazing properties are (with my TeX markup) in John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \
on “Probability (1963)” ↗…the completeness property, i.e., just took it for granted that his derivatives and integrals did yield numbers. Some of what is true about the reals is bizarre, nearly beyond belief, e.g., as in Oxtoby, Measure and Category. The main reason for being careful with fine details going from the natural numbers to the integers, rationals, algebraics, and reals is a theme that started near 1900 of trying to clean up the logical basement…
on “Trapping A Transcendental - Finding numbers that aren't solutions to polynomials” ↗…compared with the rationals, algebraics, etc. Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in John C. Oxtoby, Measure and Category. and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing! The OP wants to say that by mentioning Dedekind…
on “What are the 'real numbers', really?” ↗