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I have completed a formal proof of D&K Theorem 6.2.8 Part (ii) ... but I am unsure of whether the proof is correct ... so I would be most grateful if someone could check the proof and point out any errors or shortcomings ...

Theorem 6.2.8 reads as follows:

We have to show:

\(\displaystyle f \leq g \Longrightarrow I(f) \leq I(g) \)

where

\(\displaystyle I(f) = \int_{ \mathbb{R^n} } f(x) dx = \int_{B} f(x) dx \)

and

\(\displaystyle I(g) = \int_{ \mathbb{R^n} } g(x) dx = \int_{B} g(x) dx \)

Given a partition \(\displaystyle \mathscr{B} \text{ of } B \) we have

\(\displaystyle \overline{S} (f, \mathscr{B} ) \ = \ \sum_{j \in J} \ \sup_{ x \in B_j} \ f(x) dx \text{ vol}_n (B_j) \)

and

\(\displaystyle \overline{S} (g, \mathscr{B} ) \ = \ \sum_{j \in J} \ \sup_{ x \in B_j} \ g(x) dx \text{ vol}_n (B_j) \)

Now, since \(\displaystyle f(x) \leq g(x) \ \ for all x \in B \) we have

\(\displaystyle \overline{S} (f, \mathscr{B} ) \ \leq \ \ \overline{S} (g, \mathscr{B} ) \)

it follows that:

\(\displaystyle \overline{ \int_{B} } f(x) dx = \text{ inf} \ \{ \overline{S} (f, \mathscr{B} ) \ \ | \ \ \mathscr{B} \text{ is a partition of } B \} \ \ \leq \ \ \text{ inf} \ \{ \overline{S} (g, \mathscr{B} ) \ \ | \ \ \mathscr{B} \text{ is a partition of } B \} \ = \ \overline{ \int_{B} } g(x) dx \) ... ... ... ... ... (1)

Similarly ... ... \(\displaystyle \underline{ \int_{B} } f(x) dx \leq \underline{ \int_{B} } g(x) dx \) ... ... ... ... ... (2)

(1), (2) \(\displaystyle \Longrightarrow \int_{B} f(x) dx \leq \int_{B} g(x) dx \) ... ...

... ... that is ... ...

\(\displaystyle I(f) \leq I(g) \)

Could someone please check my proof for correctness ... and point out any errors, shortcomings and areas needing improvement ...

Peter

It may help readers of the above post to have access to D&K Section 6.2 so as to be able to check the way the authors present the theory and also to check notation so I am providing a scan of section 6.2 as it reads before Theorem 6.2.8 ... see below

Hope that helps ...

Peter

Theorem 6.2.8 reads as follows:

__Attempted Proof of Theorem 6.2.8 Part (ii)__We have to show:

\(\displaystyle f \leq g \Longrightarrow I(f) \leq I(g) \)

where

\(\displaystyle I(f) = \int_{ \mathbb{R^n} } f(x) dx = \int_{B} f(x) dx \)

and

\(\displaystyle I(g) = \int_{ \mathbb{R^n} } g(x) dx = \int_{B} g(x) dx \)

**Proof:**Given a partition \(\displaystyle \mathscr{B} \text{ of } B \) we have

\(\displaystyle \overline{S} (f, \mathscr{B} ) \ = \ \sum_{j \in J} \ \sup_{ x \in B_j} \ f(x) dx \text{ vol}_n (B_j) \)

and

\(\displaystyle \overline{S} (g, \mathscr{B} ) \ = \ \sum_{j \in J} \ \sup_{ x \in B_j} \ g(x) dx \text{ vol}_n (B_j) \)

Now, since \(\displaystyle f(x) \leq g(x) \ \ for all x \in B \) we have

\(\displaystyle \overline{S} (f, \mathscr{B} ) \ \leq \ \ \overline{S} (g, \mathscr{B} ) \)

it follows that:

\(\displaystyle \overline{ \int_{B} } f(x) dx = \text{ inf} \ \{ \overline{S} (f, \mathscr{B} ) \ \ | \ \ \mathscr{B} \text{ is a partition of } B \} \ \ \leq \ \ \text{ inf} \ \{ \overline{S} (g, \mathscr{B} ) \ \ | \ \ \mathscr{B} \text{ is a partition of } B \} \ = \ \overline{ \int_{B} } g(x) dx \) ... ... ... ... ... (1)

Similarly ... ... \(\displaystyle \underline{ \int_{B} } f(x) dx \leq \underline{ \int_{B} } g(x) dx \) ... ... ... ... ... (2)

(1), (2) \(\displaystyle \Longrightarrow \int_{B} f(x) dx \leq \int_{B} g(x) dx \) ... ...

... ... that is ... ...

\(\displaystyle I(f) \leq I(g) \)

Could someone please check my proof for correctness ... and point out any errors, shortcomings and areas needing improvement ...

Peter

__NOTE:__It may help readers of the above post to have access to D&K Section 6.2 so as to be able to check the way the authors present the theory and also to check notation so I am providing a scan of section 6.2 as it reads before Theorem 6.2.8 ... see below

Hope that helps ...

Peter

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