What is Arzela's Lemma in Fichtengoltz's Book?

  • Context: MHB 
  • Thread starter Thread starter Also sprach Zarathustra
  • Start date Start date
  • Tags Tags
    Russian
Click For Summary
SUMMARY

The discussion focuses on Arzela's Lemma as presented in Fichtengoltz's book, specifically addressing the conditions under which a point belongs to an infinite number of systems of intervals. It clarifies that the lemma states if the sum of the lengths of non-overlapping closed intervals in a finite interval [a, b] exceeds a constant positive number δ, then at least one point x = c exists in infinitely many systems D_k. Additionally, the translation of "не налегаюшие друг на друга промежутки" is confirmed as "non-overlapping intervals," emphasizing the distinction between overlapping and non-overlapping intervals in the context of the proof.

PREREQUISITES
  • Understanding of real analysis concepts, particularly interval systems.
  • Familiarity with the terminology of closed intervals and their properties.
  • Knowledge of the principles of measure theory related to interval lengths.
  • Basic proficiency in Russian for accurate translation of mathematical texts.
NEXT STEPS
  • Study the formal proof of Arzela's Lemma in Fichtengoltz's book.
  • Explore the implications of non-overlapping intervals in real analysis.
  • Learn about the applications of Arzela's Lemma in functional analysis.
  • Review measure theory concepts related to interval lengths and their sums.
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in the applications of interval systems in mathematical proofs and theorems.

Also sprach Zarathustra
Messages
43
Reaction score
0
Hello!

I've a problem understanding the following lines(Arzela lemma, and first two sentences of a proof) from Fichtengoltz's book.

I know, that some(2 members?) of you know Russian, help me please translate these line into English, with a short explanation on bold lines.

Пусть конечном промежутке $[a,b]$ содержатся системы $D_1,D_2,...,D_k,...$ промежутков, каждая из которых состоит из конечного числа не налегаюших друг на друга замкнутых промежутков. Если сумма длин промежутков каждой системы $D_k$ $(k=1,2,3,...)$ больше некторого постояного положительного числа $\delta$, то найдется, по крайней мере, одна точка $x=c$, принадлежащая бесконечному множеству систем $D_k$

Доказательство:
Если промежуток какой-нибудь системы $D_k$ $(k>1)$ налегает на промежутки предшествующих систем $D_1,D_2,...,D_{k-1}$и их концами делится на части, то эти части мы впредь будем расматривать как отдельные промежутки системы $D_k$
Thank you!
 
Physics news on Phys.org
Let the finite interval [a, b] contain the systems (i.e., sets) $D_1,D_2,\dots,D_k,\dots$ of intervals, each of which consists of a finite number of non-overlapping closed intervals. If the sum of interval lengths of each system $D_k$ ($k=1,2,3,\dots$) is greater than some fixed positive number $\delta$, then there exists at least one point x = c that belongs to an infinite set (i.e., number) of systems $D_k$.

Proof:
If an interval of some system $D_k$ (k > 1) overlaps with intervals from the preceding systems $D_1,D_2,\dots,D_{k-1}$ and is divided into parts by their ends, then we will consider these parts as separate intervals of the system $D_k$. (End of translation.)

At first I thought that "не налегаюшие друг на друга промежутки" means "disjoint intervals," but probably the right translation is "non-overlapping intervals." In the proof, when a point (the end of an interval from a preceding system) divides an interval from $D_k$ into two parts, both parts are considered elements of $D_k$. But since $D_k$ by assumption contains closed intervals, the two parts will not be disjoint, but will not "налегать друг на друга," i.e., will not overlap.