Measure theory
by Paul R. Halmos · first published 1950
10mentions
2recommended by
4HN commenters
2010–2018first – last mentioned
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…Intuitively, a measure assigns to each event A in script F a non-negative real number P(A). A signed measure permits negative values. And of course at times want a measure that yields complex numbers. Measure theory is in, say, P. Halmos, Measure Theory. The back of Halmos has a really nice introduction to probability theory, the Kolmogorov three series theorem in stochastic processes, etc. For more in stochastic…
on “Foundations of probability theory” ↗Halmos Measure Theory is a good prerequisite to Neveu. Rudin, Principles is a bit too little. Instead, the first half, the real half of Rudin's Real and Complex Analysis is a good prerequisite. So, is Royden's Real Analysis.…
on “How to learn data science” ↗Rosenlicht. Linear Algbera done Right. Then Finite Dimensional Vector Spaces. Halmos' Measure Theory. Awesome.
on “Ask HN: Learning advanced math” ↗…TeX, that σ^2 = \int (x - E[X])^2 dF_X(x) This integral need not be in the sense of Riemann (i.e., freshman calculus) because dF_X is a measure on the real line; so, the integral is in the sense of measure theory (see any of Rudin, Real and Complex Analysis; Royden, Real Analysis; Halmos, Measure Theory; Loève, Probability Theory). Sigma is defined for any random variable X or its distribution provided that E[X]…
on “Gravity waves from Big Bang detected” ↗…Then in stochastic processes I got some more respect from work of J. Doob (prof of Halmos who then became an assistant to von Neumann, etc.) and the material on probability in the back of the Halmos, Measure Theory. The Halmos work on sufficient statistics and unbiased estimation also deserved respect. I've seen no such quality from ML. You suggest that there is such quality out there, but what I saw, e.g., Ng's…
on “Acquisitions accelerate as tech giants seek to build AI smarts” ↗…control, I was interested in measurable selection and regular conditional distributions, and I got those from Breiman -- thanks Leo!!!! Yes, IIRC, there is the relevant Sierpinski exercise in Halmos, Measure Theory and also in Loeve, Probability, but it was Breiman who gave me what I really needed. (5) Generally Breiman is easy to read. He writes like he is trying to help the reader learn.
on “Foundations Machine Learning” ↗…Real and Complex Analysis. Of course, emphasize the Radon-Nikodym theorem; I like the easy steps in Royden and Loeve (below), but see also the von Neumann proof in Papa Rudin. For probability based on measure theory and the limit theorems, Breiman, Probability. Note his result on regular conditional probabilities. Neveu, Mathematical Foundations of the Calculus of Probability. If you can work all the Neveu exercises,…
on “Ask HN: Learning advanced math” ↗…'probability space' is a measurable space and a probability measure on that space, and a 'probability measure' is a non-negative measure with total mass 1. For a 'measure', see any of, say, Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ. Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York. H. L. Royden, 'Real Analysis: Second Edition', Macmillan, New York.…
on “Advanced Computer Science Courses” ↗…Interchangeability, Martingales', ISBN 0-387-90331-3, Springer-Verlag. Essentially what is going on is that we are defining P as a non-negative real 'measure' with 'total mass 1' as in any of: Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc. H. L. Royden, 'Real Analysis: Second Edition', Macmillan. Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill. This approach to probability is…
on “Think Stats, using python to learn stats ” ↗…Halmos was a student of J. Doob at University of Illinois as in: J. L. Doob, 'Stochastic Processes', John Wiley and Sons, New York, 1953. and has a very nice start on probability in: Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ, 1950. Halmos also wrote: Paul R. Halmos, "The Theory of Unbiased Estimation", 'Annals of Mathematical Statistics', Volume 17, Number 1, pages 34-43, 1946.…
on “Teachers write their own online textbook, save district $175,000” ↗