An Aspect of Lemma 3.3 .... Laczkovich and Vera T Sos .... L&S ....

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In summary: R,n) = (n^{-p})(q_1-p_1+1)(n^{-p})(q_2-p_2+1)...(n^{-p})(q_p-p_p+1)$$This is equivalent to the product notation in the statement of Lemma 3.3, proving that the upper Jordan measure of $R$ is indeed equal to the product of the lengths of the intervals in the partition $n$ that intersect with $R$.In summary, Lemma 3.3 states that the upper Jordan measure of a set $R$ is equal to the product of the lengths of the intervals in the partition $n$ that intersect with $R$. This can be proven by using the definition
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I am reading Chapter 3: Jordan Measure ... of Miklos Laczkovich and Vera T Sos's book "Real Analysis: Series, Functions of Several Variables, and Applications" (Springer) ...

I need help with an aspect or step of the proof of Lemma 3.3 ... ...

The statement and proof of Lemma 3.3 of L&S reads as follows:
L&S ... Lemma 3.3 ... ...  PART 1 ... .png

L&S ... Lemma 3.3 ... ...  PART 2 ... ... .png

All the necessary definitions together with an explanation of the notation of Lemma 3.3 are given in the scan below ...
In the proof of Lemma 3.3 we read the following:

" ... ... Therefore\(\displaystyle \overline{ \mu } (R,n) = n^{ -p} \prod_{ j =1}^p (q_j - p_j + 1 ) \) ... ... ... "Can someone please help me to prove this ... as yet i have been unable to make a meaningful start on this ...
Help will be appreciated ...

Peter
NOTE:

To make sense of Lemma 3.3 readers of the above post will need access to pages 95-97 of L&S so I providing this text as follows:
L&S ... Jordan Measure ... Ch. 3 ... Page 95 .png

L&S ... Jordan Measure ... Ch. 3 ... Page 96 ... ... .png

L&S ... Jordan Measure ... Ch. 3 ... Page 97 ... ... .png


Hope that helps ...Peter
 
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, I am happy to assist you with understanding and proving Lemma 3.3. Let's start by breaking down the statement and notation used in the proof.

First, we have the notation of $\overline{\mu}(R,n)$ which represents the upper Jordan measure of the set $R$ with respect to the partition $n$. This is defined as the sum of the volumes of the subrectangles in the partition $n$ that intersect with the set $R$, where each subrectangle has a volume of $n^{-p}$.

Next, we have the product notation, which represents the product of $p$ terms. In this case, the terms are $(q_j - p_j + 1)$, where $q_j$ represents the upper bound of the $j$th interval in the partition $n$ and $p_j$ represents the lower bound of the same interval.

Now, let's look at the statement of Lemma 3.3. It states that the upper Jordan measure of a set $R$ is equal to the product of the lengths of the intervals in the partition $n$ that intersect with $R$. This can be seen by breaking down the product notation as follows:

$$\prod_{j=1}^p(q_j-p_j+1) = (q_1-p_1+1)\cdot(q_2-p_2+1)\cdot...\cdot(q_p-p_p+1)$$

Each term in this product represents the length of an interval in the partition $n$, and when multiplied together, they give us the total volume of the subrectangles in the partition $n$ that intersect with $R$. Multiplying this by $n^{-p}$ gives us the total volume of the subrectangles, which is equal to the upper Jordan measure of $R$.

To prove this statement, we can use the definition of upper Jordan measure and the properties of products. We know that the upper Jordan measure of a set $R$ is equal to the sum of the volumes of the subrectangles in the partition $n$ that intersect with $R$. Using the definition of volume, we can write this as:

$$\overline{\mu}(R,n) = \sum_{j=1}^p(n^{-p})(q_j-p_j+1)$$

We can now use the properties of products to rewrite this as a product, as follows:

$$\overline{\
 

1. What is Lemma 3.3 in the paper by Laczkovich and Sos?

Lemma 3.3 in the paper by Laczkovich and Sos is a mathematical statement that provides a necessary condition for a set to have a certain property. It is a part of a larger proof in the paper.

2. What is the significance of Lemma 3.3 in the paper?

The significance of Lemma 3.3 lies in its role in the overall proof presented in the paper. It helps to establish a necessary condition for a set to have a certain property, which is crucial in understanding the main result of the paper.

3. Can you explain the proof of Lemma 3.3 in simple terms?

The proof of Lemma 3.3 involves using mathematical techniques and logic to show that a set must have a certain property in order for the main result of the paper to hold true. It may be difficult to explain the proof in simple terms without a background in mathematics.

4. How does Lemma 3.3 relate to the rest of the paper?

Lemma 3.3 is an important part of the overall proof presented in the paper. It helps to establish a necessary condition for a set to have a certain property, which is crucial in understanding the main result of the paper. Without Lemma 3.3, the proof would not be complete.

5. Is Lemma 3.3 applicable in other areas of mathematics?

Yes, Lemma 3.3 can be applied in other areas of mathematics where similar properties are being studied. It provides a general framework for understanding necessary conditions for certain properties to hold true in different contexts.

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