Measure Theory and Functional Analysis
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PhD thesis, University of Trento. This thesis is concerned with the study of nonstandard models in measure theory and in functional analysis. In measure theory, we define elementary numerosities, that are additive measures that take on values in a non-archimedean field and for which the measure of every singleton is 1. We have shown that, by taking the ratio with a suitable unit of measurement, from a numerosity it can be defined a non-atomic real-valued measure, and that every non-atomic measure can be obtained from a numerosity by this procedure.
We then used numerosities to develop a model for the probability of infinite sequences of coin tosses coherent with the original ideas of Laplace.
Nonstandard Models in Measure Theory and in functional Analysis
In functional analysis, we introduce a space of functions of nonstandard analysis with a formally finite domain, that extends both the space of distributions and the space of Young measures. As it turns out, this is the only thing that goes wrong with the DCT for Riemann integrable functions.
To be precise, there is DCT appropriate for Riemann integrals you just throw in the assumption that the limit function is also Riemann integrable — an assumption that holds in practice many times , that can be taught to first year math majors. Mostly for my own purposes, I hope to write more on this soon.
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To sum up, measure theory is indeed a must, but so are many other subjects. So it is good to think how to maneuver when your students did not learn it. Much can still be done!
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Another case for measure theory is that you need it in order to do probability right. Not long ago, I had this very same debate about measure theory with another graduate student at our department. And I do agree with everything you wrote here, apart from the last brackets concerning probability. Actually I think the answer lies exactly there. After a month we define an expectation as follows: — If is a simple random variable is what you think it should be; — If is non-negative random variable is the supremum of where runs over all simple random variables with ; — If is any random variable , unless it is.
In this way, when one teaches functional analysis, one CAN give as an example of a Hilbert space they even know what is the inner product and many other examples. Thanks for the comment.
Math - Measure Theory and Functional Analysis I - Fall
Unfortunately, this is sometimes very difficult to do. Reblogged this on Being simple. I heard nice opinions of it.
Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must.
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Of course anyone going for a graduate degree in math should study measure theory and get an A , […]. It is different in different courses, in different programs and different schools. I taught once a course in Fourier analysis for engineers which clearly had not functional analysis or measure theory prerequisite, though some of the methods or modes of thinking of functional analysis were introduced along the way. You are commenting using your WordPress.
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Math 6312 Measure theory
Notify me of new posts via email. The introduction kind of got out hand, so I decided to post it separately.
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Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break but in any case before we need them for the course I am teaching this term, Calculus 2 ].
But… Take for example the DCT. Share this: Twitter Facebook. Like this: Like Loading Published: March 26, Filed Under: Academia , Calculus , teaching , Thoughts on mathematics.