Anyone here know anything about measure theory? Can someone explain it to me? How do I conceptualize a set of measure zero?

Also, mathematicians enjoy using the word "moral" a lot -- what the heck does it mean to say that "Thus, morally, a square-integrable function is equal to its Fourier series"? I know in graph theory "moralizing" a graph means to join ('wed') the two parents of a child node. Mathematicians must have horribly depressing personal lives, which is why they resort to such abstraction.

Presumably the whole point of this argument is to show how a square integrable function (i.e., one whose Lebesgue integral converges) can be decomposed into a Fourier series, and by arguments of measure theory any arbitrary function can be described by simpler square-integrable functions defined over a complete complex inner-product space (Hilbert space).

Anyone who can cut through all this and give me the skinny?

oh yeah... the context of all this is Fourier decompositions.

I view the necessity for the convergence of square integrable functions as simply an argument that says that all approximatable functions must have a finite autocovariance function, |f(x)|^2 (and in this case, with respect to the phase angles, x, so they must have a finite auto-coherence).

Recall, a square-integrable function is as follows:

integral [-pi,pi] of |f(x)|^2 dx

which must be less than infinity if it is to converge. Also, f(x) is defined on a complex space (a + ib).

Thanks.

Hi GundamWing,

This is the first time I hear of the word "moral" in mathematics... I won't be able to help you on that, but I can say something about square integrable functions (let's call all these functions L² from now on).

"Presumably the whole point of this argument is to show how a square integrable function (i.e., one whose Lebesgue integral converges) can be decomposed into a Fourier series, and by arguments of measure theory any arbitrary function can be described by simpler square-integrable functions defined over a complete complex inner-product space (Hilbert space). "

I won't go into detail of the proofs you mention here, but I can attempt to explain what it means.

First of all, you should know that all the maths you mention here are used in quantum mechanics. I will use that picture to make things clearer (hey, I am a physicist, not a mathematician, thus I am allowed to do that ).

You establish that any L² function can be decomposed in a Fourier series; this basically means that if you know a wavefunction W(x), then you can also calculate its associated momentum-function W'(p), which is simply the Fouriertransformed (closely related to Fourier series, as a matter of fact the L² functions are also Fourier-transformable). This is a mathematical formulation of the physical intuition: "If we know there is a particle with wavefunction W(x) then we also know it has some momentum that we can express in some function W'(p)".

Then you talk about Hilbertspaces and how any arbitrarily function can be expressed in L² functions. This is the first time I hear this result, but I'll assume it is correct. Then what does it mean ? Quite simple: if you take a function f(x), and you look at all (lineary independent) L²-functions q_i(x), then you can find coefficients c_i such that:

f(x) = <font face="symbol">S</font> c_i q_i(x)

up to any given precision (that is why you need a complete space, it eases the maths to express that "something is arbitrarily close to something else" or "up to any given precision they are the same"). The coefficients c_i are defined as the inner product of f(x) and q_i(x) in that Hilbertspace.

Compare this to ordinary vector calculus: you have three vectors (1,0,0) and (0,1,0) and (0,0,1) in which all the other vectors (a,b,c) can be expressed. Denote e_i one of those three vectors, then:

(a,b,c) = <font face="symbol">S</font> c_i e_i

where the c_i are a,b,c respectively (note by the way that c₁ a = scalar/inner product of (a,b,c) and e_i(1,0,0), c₂ = scalar/inner product of (a,b,c) and e₂, ... )

Hope this helps a bit,

Crisp

Crisp gave a very simple, clear explanation of Lebesque measure so I won'r repeat that.


I just want to note that I also have NEVER seen the word "moral" used in mathematics (I do remember a joke about
eigenvalues have a moral resonsibility to be "discrete" but that was Physics!).

Yet, GundamWing says mathematicians use it a lot!

I don't know what your native language is (and on this forum, I wouldn't want to guess!) but are you sure that "moral" isn't a misinterpretation of a word?

Thanks guys. I appreciate the re-capitulation in terms of physics, that is more intuitive.

I'm quite sure "moral" is not a misninterpretation, because i'm reading it in english, and that's exactly how it is spelled. Look up stuff on Graphical Models (the word "moral" turns up a lot! and in fact it implies the 'connection between parent nodes' of a 'child' node in the language of graphical tree structured models).

check the following out --
>> http://www.cs.technion.ac.il/~cs236372/tirgul05.pdf

A triangulated graph is called 'moral'.

I did find that a "set of measure zero" can be taken to mean for example certain discontinuities in functions (where you have two values of the function f for a particular x).

Lebesgue functions can apparently be constructed by taking the limit of simple measurable functions that approximate the true function which is equivalent to the Riemann integral.

One thing that isn't so clear is why should the integral of any measurable function over the rationals be equal to zero? Doesn't this imply that the function is 'symmetric' over the rationals? Otherwise, it doesn't make sense.

Hi GundamWing,

"One thing that isn't so clear is why should the integral of any measurable function over the rationals be equal to zero? Doesn't this imply that the function is 'symmetric' over the rationals? Otherwise, it doesn't make sense."

Ehr, no... the Lesbegue integral "is defined" such that the integral over a neglegible set (in this case, the rationals in the real numbers) is zero. This makes it a bit more natural from a measure-theory point of view... Neglegible means exactable that the "influence of the set is almost undetectable", so somehow you would want neglegible sets to give no contribution in integrals.

Bye!

Crisp