Some Questions about Papa Rudin (RCA) Chapter 1

[joseph.hu37]

Hi dear friends over the Internet,

I have some questions on Papa Rudin:

Question 1:

On page 12 (proof of Theorem 1.9e), why is it that E is measurable?

Question 2:

On page the bottom of page 15 (the proof of Theorem 1.17), why are the φn's Borel functions? Also, the proposition states that f is a measurable function from X to [0,∞ ], but doesn't the definition (at least given in Def 1.3 on p8) require [0, ∞] being a topological space? Then what would be the topology of [0, ∞]?

Thanks, any help is greatly appreciated.

 

[jgens]

Question 1:

On page 12 (proof of Theorem 1.9e), why is it that E is measurable?

Notice that |f| is real measurable and that X\E=|f|^-1((0,∞)).

Question 2:

On page the bottom of page 15 (the proof of Theorem 1.17), why are the φn's Borel functions? Also, the proposition states that f is a measurable function from X to [0,∞ ], but doesn't the definition (at least given in Def 1.3 on p8) require [0, ∞] being a topological space? Then what would be the topology of [0, ∞]?

The topology is generated by the open intervals and intervals of the forms [0,b) and (a,∞]. To prove that φ_n is a Borel mapping, let U be an open set in [0,∞]. If kδ_n∈U, then [kδ_n,(k+1)δ_n)⊂φ_n^-1(U). If n∈U, then [n,∞]⊂φ_n^-1(U). It follows from this that φ_n^-1(U) is a union of Borel sets and therefore φ_n is a Borel mapping.

 

[joseph.hu37]

Hi jgens, thanks for your answer. You are absolutely amazing! Thanks.