Measure Theory

[Xittenn]

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?

[Bignose]

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

[Xittenn]

Kind of like how ajb has mentioned about Grassman Numbers and Green's Theorem/Function?

[ajb]

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.

[Xittenn]

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.

[D H]

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.

[ajb]

Measure theory is such an important concept.

 

Indeed, measure theory is an important notion in quantum theory, though often very formal.

[Xittenn]

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.