Function required to be integral - (measure theory?)

[jbusc]

Hi,

Some time ago one of my professors told us about a remarkable theorem, which stated something along the lines of: if a function i takes two arguments, one being another function f, and the other being some region R on which the function f is defined, and this function i satisfies some particular properties (such as linearity in f, if disjoint regions st R1 U R2 = R then i(f, R1) + i(f, R2) = i(f, R), etc) which are similar to the integral, then the function i _must_ be the integral of f.

I later asked him again, and he said it was a basic result in measure theory (which he did not pursue further since it was not a measure theory class). However I have been unable to find this theorem again, partly because I never was quite sure of the exact properties the theorem requires, and also because I never pursued more measure theory. I would really appreciate it if someone could recognize this theorem and either point me to a source, clarify the conditions, or even just tell me what it's called.

Thanks!

 

[HallsofIvy]

Yes, if a functional has the properties that it is linear in f, "additive", "translation invarient", and such that if f(x)= 1 for x in R then the functional returns the area of R, it must be the integral.

Check out
http://en.wikipedia.org/wiki/Measure_theory

or goggle on '"Lebesgue measure" uniqueness'

 

[tacman]

Hello,

The theorem you are referring to is known as the Lebesgue's Dominated Convergence Theorem, and it is indeed a fundamental result in measure theory. It states that if a function i satisfies the properties you mentioned (linearity, additivity, etc.) and also satisfies the condition that it is dominated by a measurable function g on the region R, then i must be the integral of f over that region R.

This theorem is important because it allows us to interchange the order of integration and limit in certain cases, which is crucial in many areas of mathematics and physics. It also provides a way to evaluate integrals that may otherwise be difficult to compute directly.

I hope this helps clarify the conditions and name of the theorem for you. If you would like to learn more about measure theory and its applications, I would suggest looking into textbooks such as "Measure Theory and Integration" by G. de Barra or "Real Analysis" by H.L. Royden. Good luck with your studies!