Proving that the limiting function is in Lp

  • Thread starter bbkrsen585
  • Start date
  • Tags
    Function
In summary, To show that the pointwise limit g is in Lp, we need to show that the integral of |g|^p is finite. This can be proven using Fatou's lemma by showing that the sequence of |g_n|^p converges to |g|^p almost everywhere. This follows from the continuity of |~|^p.
  • #1
bbkrsen585
11
0

Homework Statement



Hi guys. I have one question regarding convergence in Lp. Suppose gn converges to g mu-almost everywhere on [0,1]. Suppose further that [tex]\left\|[/tex]gn[tex]\left\|[/tex]p[tex]\rightarrow[/tex]M < [tex]\infty[/tex]. How do I show that the pointwise limit g is in Lp?

Thanks.

Homework Equations



So far, I know this is not true if we only know that gn goes to g mu-almost everywhere. I just don't see how the additional condition that the Lp-norms converge to some finite number implies that the limiting function g is also in Lp.
 
Physics news on Phys.org
  • #2
So, you need to show that

[tex]\int |g|^p<+\infty[/tex].

Since [tex]|g_n|^p\rightarrow |g|^p[/tex], we need to show

[tex]\int \lim_{n\rightarrow +\infty}{|g_n|^p}<+\infty[/tex]

Now, apply Fatou's lemma...
 
  • #3
But, how do we know that |gn|^p converges to |g|^p a.e.?

If gn goes to g almost everywhere does that imply that |gn|^p converges to |g|^p a.e.? That's not so clear to me. Thanks.
 
  • #4
It follows from the continuity of [tex]|~|^p[/tex].
 

What is the definition of Lp in mathematics?

In mathematics, Lp is a function space that consists of all p-power integrable functions on a given measure space. It is commonly used to measure the size or complexity of a function.

What is the importance of proving that the limiting function is in Lp?

Proving that the limiting function is in Lp is important because it allows us to determine the convergence and behavior of a sequence of functions. It also helps us to study the properties of a function and its relationship with other functions in the Lp space.

What is the process of proving that the limiting function is in Lp?

The process of proving that the limiting function is in Lp involves using mathematical techniques such as the Cauchy-Schwarz inequality and the Hölder's inequality to show that the sequence of functions converges to a function that is in the Lp space.

What are some commonly used techniques for proving that the limiting function is in Lp?

Some commonly used techniques for proving that the limiting function is in Lp include using the dominated convergence theorem, the monotone convergence theorem, and the Fatou's lemma. These techniques help to simplify the proof and provide a rigorous justification for the convergence of the sequence of functions.

What are the practical applications of proving that the limiting function is in Lp?

The practical applications of proving that the limiting function is in Lp include solving problems in areas such as probability, statistics, signal processing, and functional analysis. It also plays a crucial role in the development of various mathematical models and algorithms used in engineering, physics, and other scientific fields.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
225
  • Calculus and Beyond Homework Help
Replies
2
Views
782
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
128
  • Calculus and Beyond Homework Help
Replies
4
Views
5K
Replies
0
Views
275
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
Replies
23
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
922
Replies
7
Views
1K
Back
Top