Why does an open set in Riesz-Nagy's theorem decompose into intervals?

[Castilla]

Good Morning.

I am reading the first pages of the "Lessons of Functional Analysis" of Riesz and Nagy, because I learned that this book was the main source for Apostol chapter about the Lebesgue Integral, and I am trying to find a proof of the Fundamental Theorem of Calculus for Lebesgue Integrals worked out within this approach (not Measure Theory).

One of the first theorems of Riesz-Nagy's book states: "Every monotonic function f(x) posseses a finite derivative at every point x with the possible exception of the points x of a set of measure zero".

To proof this T they use this lemma:

"Let g(x) be a continuous function defined in the closed interval (a, b), and let E be the set of points x interior to this interval and such that there exists and e lying to the right of x with g(e) > g(x). Then the set E is either empty or an open set, i.e., it decomposes into a finite number or a denumerable infinity of open and disjoint intervals (a_k, b_k) (the underline is mine), and g(a_k) < g(b_k) for all these intervals".

Well I suppose I can follow the lemma's proof on the book but in this
statement there is something I did not know. Why to be E an open set implies that it decomposes into a finite number or a denumerable infinity of open and disjoint intervals ?

 

[WWGD]

Good Morning.

I am reading the first pages of the "Lessons of Functional Analysis" of Riesz and Nagy, because I learned that this book was the main source for Apostol chapter about the Lebesgue Integral, and I am trying to find a proof of the Fundamental Theorem of Calculus for Lebesgue Integrals worked out within this approach (not Measure Theory).

One of the first theorems of Riesz-Nagy's book states: "Every monotonic function f(x) posseses a finite derivative at every point x with the possible exception of the points x of a set of measure zero".

To proof this T they use this lemma:

"Let g(x) be a continuous function defined in the closed interval (a, b), and let E be the set of points x interior to this interval and such that there exists and e lying to the right of x with g(e) > g(x). Then the set E is either empty or an open set, i.e., it decomposes into a finite number or a denumerable infinity of open and disjoint intervals (a_k, b_k) (the underline is mine), and g(a_k) < g(b_k) for all these intervals".

Well I suppose I can follow the lemma's proof on the book but in this
statement there is something I did not know. Why to be E an open set implies that it decomposes into a finite number or a denumerable infinity of open and disjoint intervals ?

Every open set in the Real line is the countable union of open intervals. You may argue component-wise. If the set is connected it must be an interval. Induct on the number of components. You will have a rational in each component, this gives you the countability.