How Do Apostol's and Le's Definitions of Measurable Functions Relate?

[Castilla]

First I had learned this definition of a "measurable function" (Apostol):

"Let I be an interval. A function f: I -> R is said to be measurable on I if there exists a sequence of step functions (s_n(x) ) such that s_n(x) -> f(x) as n -> infinity for almost all x in I."

But now in other sources (for example, the web notes of proffesor Dung Le) I have found this definition: "A function f: E -> R is measurable if E is a measurable set and for each real number r the set { x in E / f(x) > r } is measurable."

First I thought Apostol and Le were talking about different things, but afterwards I found that both use their definitions to prove similar theorems, as this one:

(In Apostol version):
"Suppose that I is an interval and that f, g are measurable functions on I. Then so are f + g, f - g, fg, |f|, max {f,g}, min {f,g}".

(In Dung Le version):
"Let f: E-> R and g: E -> R be measurable functions. Then the functions (k is a real) kf, f+ g, |f| and fg are measurable."

I was trying to learn the Fundamental Theorem of Calculus with Lebesgue integrals. Dung Le has it in his notes, but he uses the second definition of a measurable function and I have learned the basics of Lebesgue integration in Apostol, which only uses the first definition.

Furthermore, to proof the FTC I see that Dung Le use a lemma by which "the function equivalent to the infimun of a set of measurable functions is also measurable". But I don't know if this lemma has an equivalent in the Apostol approach and with Apostol's definition of a measurable function.

Is there is some way of reunite these two definitions in one?

Thanks for your answers.

 

[matt grime]

I'm pretty sure that these are equivalent definitions. Measurable is such a universal term I do not believe there would be two inequivalent definitions for it.Certainly Apostols' is equivalent to the second (step functions are measurable in the second sense, and hence since measurable functions are closed under limits, anything in Apostol's definition satisfies Le's), and the converse is to prove that step functions are dense in Le's definition which seems clear if messy: split the range up into intervals, inverse image of each is measurable, approximate with step functions which we can therefore do, yeah, ought to work.

 

[Castilla]

Thanks four answering, Matt.

My "problem" is that I have learn some things about Lebesgue integration in Apostol's book but now it seems that to understand the Fundamental Theorem of Calculus for Lebesgue Integrals what I have learned helps little or nothing and I have to begin from zero with arid measure theory.

1. In the Apostol approach, the Lebesgue integral on an interval I is defined first for step functions (in that case is the same that the Riemann integral).

2. Then we define the "upper functions". f is an upper function iff:

- there exists a sequence of increasing step functions (s_n(x)) which converges to f(x) almost everwhere on I.

- there exists lim (n->oo) integral s_n(x)dx, and we define
integral f(x)dx = said limit.

3. Then we define Lebesgue functions this way: v is a Lebesgue function if there exists two upper functions f, g such that v = f - g.

We learn here the monotone convergente theorem, the beppo levi's theorem and the Lebesgue dominated theorem.

4. Then we define measurable functions (see Apostol's definition on my first post) and we learned, among others, the "diferentiation under the integral sign" theorem for lebesgue integrals.

All of this with little or nothing of Measure Theory.

But now I want to learn the FTC for Lebesgue integrals and in all sources I have found this theorem is embebbed on an approach completely based on dull, arid measure theory.

There must exist some book which shares Apostol approach and includes the FTC! Or not?