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## Main Question or Discussion Point

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 dont 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.

"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 dont 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.