A Question in a proof on Lebesgue integral under diffeomorphism

In summary, the conversation discusses a question from Theorem 2.47 in Folland's "real analysis: modern techniques and their applications", second edition, specifically regarding inequalities on page 76. The question involves the use of the measure property "continuity from above" and the applicability of this property depends on the finiteness of m(G(U_1)) or m(G(U_j)) for at least one of j>1. To prove this, it is necessary to show that \int_{U_1}|\det D_x G|dx is finite. Additionally, to use the dominated convergence theorem, it is required that |\det D_x G| is integrable on U_j, meaning \int_{U_j
  • #1
zzzhhh
40
1
This question comes from proof of Theorem 2.47 in Folland's "real analysis: modern techniques and their applications", second edition. In particular, the question lies in the inequalities in line 7 and 8 in page 76. The first equality is an application of measure property "continuity from above". But for this property to be applicable, [tex]m(G(U_1))[/tex] (or [tex]m(G(U_j))[/tex] for at least one of j>1) must be finite. If we can prove that [tex]\int_{U_1}|\det D_x G|dx[/tex] is finite, we can derive this result by the inequality in line 3. Secondly, to use dominated convergence theorem to obtain the last "=", [tex]|\det D_x G|[/tex] must be integrable on [tex]U_j[/tex], that is, [tex]\int_{U_j}|\det D_x G|dx[/tex] must be finite. But I can not prove this. I tried to show that [tex]|\det D_x G|[/tex] is bounded on [tex]U_j[/tex], but [tex]U_j[/tex] is only an open set, although it has finite measure. Could you please help me prove that the integral [tex]\int_{U_j}|\det D_x G|dx[/tex] is finite? Thanks!
The following image contains part of this proof, for reference.
21l7gqf.png
 
Physics news on Phys.org
  • #2
The exposition here is wrong, check the errata 1 of this book from the author's website ...
oops!
 

1. What is the Lebesgue integral?

The Lebesgue integral is a mathematical concept used to calculate the area under a curve. It was developed by French mathematician Henri Lebesgue in the early 20th century as a more general and powerful alternative to the Riemann integral.

2. What is a diffeomorphism?

A diffeomorphism is a type of function that maps one differentiable manifold onto another, preserving the smoothness of the original manifold. In simpler terms, it is a function that is both smooth and invertible.

3. How are the Lebesgue integral and diffeomorphisms related?

The Lebesgue integral is invariant under diffeomorphisms, meaning that if we apply a diffeomorphism to a function, the Lebesgue integral of the resulting function will be the same as the Lebesgue integral of the original function. This allows us to use diffeomorphisms to simplify calculations involving Lebesgue integrals.

4. What is the purpose of the question in the proof on Lebesgue integral under diffeomorphism?

The question in the proof serves to show the relationship between diffeomorphisms and Lebesgue integrals. By proving that the two are invariant under each other, we can use this knowledge to simplify and generalize our calculations involving Lebesgue integrals.

5. How is the Lebesgue integral under diffeomorphism different from the standard Lebesgue integral?

The Lebesgue integral under diffeomorphism takes into account the change of variables caused by applying a diffeomorphism to a function. This allows for more flexibility and simplification in the calculation of Lebesgue integrals, as it takes into account the transformation of the underlying space.

Similar threads

Replies
20
Views
2K
Replies
1
Views
830
Replies
7
Views
1K
Replies
0
Views
836
  • Calculus
Replies
7
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
964
Replies
4
Views
306
Replies
3
Views
2K
  • Topology and Analysis
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
792
Back
Top