Xittenn Posted April 17, 2012 Share Posted April 17, 2012 What is the importance of measure theory, it is a branch that I've heard little about? Can anyone point me at the definitive text on its foundation? Link to comment Share on other sites More sharing options...
Bignose Posted April 17, 2012 Share Posted April 17, 2012 (edited) It generalizes the concept of the integral. Typically, the integral is introduced as the area under a given curve. Area is just a single specific example of a 'measure' -- there are many others. Take the following nasty function: f(x) = 1 if x is irrational, = 0 if x is rational. This function's integral isn't defined if you limit yourself to integral = area under the curve. But, with an appropriate measure, the integral can take on meaningful values. Searching for 'Lebesgue measure' should be a good start. There are plenty of good texts, depending in your current level of mathematical understanding. I bought a book at a library sale years ago called Volume and Integral by Rogosinski for like $1. Anything with a title like Measure and Integral or similar should fit the bill. Edited April 17, 2012 by Bignose 1 Link to comment Share on other sites More sharing options...
Xittenn Posted April 17, 2012 Author Share Posted April 17, 2012 Kind of like how ajb has mentioned about Grassman Numbers and Green's Theorem/Function? Link to comment Share on other sites More sharing options...
ajb Posted April 17, 2012 Share Posted April 17, 2012 A measure [math]\mu[/math] is a law which assigns a number [math]\mu(A)[/math] to certain subsets [math]A[/math] of a given space. You should think of a measure as giving a generalisation of length of an ineterval or the probability that an event from [math]A[/math] occurs. Given a measure one can define an integral of suitable functions with respect to [math]\mu[/math]. Basically, the Riemann integral works for continuous functions. If we have functions that are discontinuous ``almost everywhere'' the Riemann integral fails, but we can use a more general notion, the Lebesgue measure, to come up with a good notion of "integral". Measure theory in general is roughly making sense of "integration" for functions more general than just the continuous ones. 1 Link to comment Share on other sites More sharing options...
Xittenn Posted April 17, 2012 Author Share Posted April 17, 2012 Thanks guys! It seems that "Real and Complex Analysis" - Rudin covers the topics well and this was already on my list so I think I'll check it out! It seems that it is generally treated at the graduate level, so I guess I'll just read through and go from there. Link to comment Share on other sites More sharing options...
D H Posted April 18, 2012 Share Posted April 18, 2012 Measure theory is such an important concept. Orthogonal polynomials, Fourier analysis, probability theory: All of a sudden these things make a whole lot more sense when looked at from the perspective of measure theory. Some mathematicians advocate for teaching Lebesgue integration from day one. BTW, Rudin is a very good place to start. 1 Link to comment Share on other sites More sharing options...
ajb Posted April 18, 2012 Share Posted April 18, 2012 Measure theory is such an important concept. Indeed, measure theory is an important notion in quantum theory, though often very formal. Link to comment Share on other sites More sharing options...
Xittenn Posted April 18, 2012 Author Share Posted April 18, 2012 It seems to be required knowledge for understanding Hilbert Spaces and Spectral Multiplicity, at least according to Halmos. I figured I would read get caught up on Measure before diving into this other topic. Link to comment Share on other sites More sharing options...
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