Unclear steps in a Zorich proof (Measurable sets and smooth mappings)

In summary, the conversation discusses the proofs of statements a), b), and c) from Zorich's Mathematical Analysis II section 11.5.2. The conversation questions why ##\overline{E}_t## in the proof of b) is compact and how its boundedness is determined. It also questions the boundedness of ##E_x=\phi(E_t)## in the proof of c), as well as the implication of having a Jordan measure of zero for ##\overline{E}_t##.
  • #1
Unconscious
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From Zorich, Mathematical Analysis II, sec. 11.5.2:

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where as one can read from the statement, the sets could also be unbounded.

I do not report here the proof of the fact a), beacuse I have no doubt about it and one can, without the presence of dark steps in the reasoning, assume a) as prooved and pass to the analysis of b) and c).
The proposed proofs of these are the following:

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My questions are:

proof of b): why is ##\overline{E}_t## compact? In particular, I can't understand its boundedness.

proof of c): by hypothesis ##E_t## is Jordan-measurable (that is, it is bounded with measure-zero boundary). Then, why also should ##E_x=\phi(E_t)## be bounded, in general?
 
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  • #2
I haven't thought about jordan measure ever before, but if ##\overline{E}_t## has jordan measure zero, does that imply it must be compact? It seems like a plausible consequence of the definition. I think if it's not bounded, then any finite set of rectangles you draw that cover the whole set must have infinite area, right?
 
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Likes Unconscious
  • #3
Right, thank you.
 

Related to Unclear steps in a Zorich proof (Measurable sets and smooth mappings)

1. What is the purpose of Zorich proofs?

The purpose of Zorich proofs is to provide a rigorous mathematical framework for understanding and analyzing measurable sets and smooth mappings. These proofs are used in various fields of mathematics, including analysis, topology, and differential geometry.

2. What makes the steps in a Zorich proof unclear?

The steps in a Zorich proof may be unclear due to the complexity of the mathematical concepts involved, the use of advanced techniques and notation, or the lack of detailed explanations or examples. Additionally, different readers may have varying levels of understanding and may find certain steps more difficult to follow than others.

3. How can one overcome difficulties with unclear steps in a Zorich proof?

One way to overcome difficulties with unclear steps in a Zorich proof is to consult additional resources, such as textbooks, lecture notes, or online tutorials, to gain a better understanding of the concepts and techniques used. It may also be helpful to discuss the proof with peers or seek guidance from a mentor or instructor.

4. Are there any tips for understanding Zorich proofs?

Some tips for understanding Zorich proofs include breaking down the proof into smaller, more manageable parts, carefully studying the definitions and notation used, and trying to relate the steps to known theorems or concepts. It may also be helpful to work through examples and practice problems to gain a better understanding of the material.

5. How important is it to understand every step in a Zorich proof?

While it is important to have a general understanding of the steps in a Zorich proof, it is not always necessary to understand every single step in great detail. It may be more important to grasp the overall concept and the key ideas behind the proof, rather than memorizing every technical detail. However, a deeper understanding of the proof can help in applying the concepts to other problems and in furthering one's understanding of the subject.

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