Sandwich theorem for lebesgue integral

In summary, the conversation discusses the existence of a sandwich theorem for Lebesgue integrals. The participants also address the related claim of whether a function sandwiched between two integrable functions is also integrable. The conversation goes on to discuss the conditions for the dominated convergence theorem and a specific result related to the convergence of measurable functions. The participants also mention possible references for further study.
  • #1
onthetopo
35
0
It would be helpful if there is a sandwich theorem for lebesgue integreal. Does it exist?
Ie. if fn<=hn<=gn on measure space (X,S,u) and fn and hn are integreable and measurable , then
g must also be in L1.

A related claim is that if fn-> f and hn->h, does gn also necessarily converge to some g?
 
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  • #2
Did you mean to ask if h_n (not g) is in L^1? What do you know about h_n - in particular, is it measurable? If it's just an arbitrary function then it's easy to come up with examples of nonmeasurable functions sandwiched between L^1 functions...

On the other hand, if h_n is measurable, then it is L^1 by the monotonicity of the integral (you have to be a little careful here).

And when you write f_n -> f, what notion of convergence are you using?

For future reference, you should always post your ideas when you ask for help. Forum rules and such.
 
  • #3
morphism said:
Did you mean to ask if h_n (not g) is in L^1? What do you know about h_n - in particular, is it measurable? If it's just an arbitrary function then it's easy to come up with examples of nonmeasurable functions sandwiched between L^1 functions...

On the other hand, if h_n is measurable, then it is L^1 by the monotonicity of the integral (you have to be a little careful here).

And when you write f_n -> f, what notion of convergence are you using?

For future reference, you should always post your ideas when you ask for help. Forum rules and such.

Sorry , you are right.
Ideally, if fn, gn, and hn are all nonnegative, we can use comparison theorem to show that hn is in L1. But how about the case that fn, gn, hn are simply any measurable functions?
 
  • #4
Any measurable function... is it integrable? If we have fn<=gn<=hn with fn and hn integrable, then |fn| and |hn| are integrable as well, and hence max(|fn|,|hn|) is also. What can you say about |gn| compared to this?
 
  • #5
The question should include fn, hn, f,h integreable
then max(|fn|,|hn|) is also integreable,
then |gn|<x(|fn|,|hn|), thus gn is in L1,
But still it doesn't say "integral gn" converges to "integral g" for some g
 
  • #6
That last bit follows from the dominated convergence theorem.
 
  • #7
To use DCT, we need to show there exists a g such that g->gn
but this is true, since |gn|<|fn|+|hn|<|f|+|h|+2epsilon
thus |gn| converges to |f|+|h|
Now, suppose the space is complete, then every absolutely convergent sequence is convergent, thus there exists a g such that gn->g
is this right?
 
  • #8
Hold on - no. If you don't know that g_n converges in the first place, then you can't say anything.

On the other hand, we do have the following result: Let {f_n} and {h_n} be sequences of integrable functions that converge a.e. to f and h (which are also integrable functions). Let {g_n} be a sequence of measurable functions such that f_n <= g_n <= h_n for all n, and such that g_n -> g pointwise. If [itex]\int f_n \to \int f[/itex] and [itex]\int h_n \to \int h[/itex], then [itex]\int g_n \to \int g[/itex].

I don't know off the top of my head if any of the conditions can be relaxed on this. (In fact, they might even require some strengthening!)
 
  • #9
morphism said:
Hold on - no. If you don't know that g_n converges in the first place, then you can't say anything.

On the other hand, we do have the following result: Let {f_n} and {h_n} be sequences of integrable functions that converge a.e. to f and h (which are also integrable functions). Let {g_n} be a sequence of measurable functions such that f_n <= g_n <= h_n for all n, and such that g_n -> g pointwise. If [itex]\int f_n \to \int f[/itex] and [itex]\int h_n \to \int h[/itex], then [itex]\int g_n \to \int g[/itex].

I don't know off the top of my head if any of the conditions can be relaxed on this. (In fact, they might even require some strengthening!)

Wonderful, How did you prove this? if too long, what reference book did you find this theorem?
 
  • #10
I did an exercise of this sort back when I took measure theory (so I might have gotten some of the details wrong). If I'm not mistaken, this might also be an exercise in Royden (in the chapter that contains the Lebesgue dominated convergence theorem for the real line; chapter 3?). Anyway, the idea is to use the dominated convergence theorem. It might also be helpful to consider the case f_n=-g_n (so |h_n| <= f_n) first.
 
  • #11
why do we need gn->g pointwise
|gn|<|fn|+|hn|<|f|+|h|+2epsilon
thus |gn| converges to |f|+|h|
 
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What is the sandwich theorem for Lebesgue integral?

The sandwich theorem for Lebesgue integral, also known as the squeezing theorem, is a fundamental result in measure theory that allows us to evaluate the integral of a function by bounding it between two other functions. It is similar to the sandwich theorem in calculus, but applies to Lebesgue integrals in the context of measure theory.

How does the sandwich theorem help in computing Lebesgue integrals?

The sandwich theorem provides a useful tool for computing Lebesgue integrals, particularly when the integrand is difficult to evaluate directly. By finding upper and lower bounds for the function, we can use the sandwich theorem to show that the integral exists and has a specific value.

What are the conditions for the sandwich theorem to hold for Lebesgue integrals?

In order for the sandwich theorem to hold for Lebesgue integrals, the two bounding functions must have the same domain and be integrable on that domain. Additionally, the integrand must be bounded between the two functions, and the integral of the difference between the two bounding functions must approach zero as the size of the interval approaches zero.

Can the sandwich theorem be applied to improper Lebesgue integrals?

Yes, the sandwich theorem can be applied to improper Lebesgue integrals, as long as the conditions for the theorem to hold are satisfied. This allows us to evaluate integrals that may not have a finite value otherwise.

What are the practical applications of the sandwich theorem in science and mathematics?

The sandwich theorem has many practical applications in science and mathematics, particularly in areas such as physics, engineering, and economics. It can be used to prove the convergence of series and improper integrals, as well as in the study of limits and continuity of functions. In addition, it provides a useful tool for approximating difficult integrals and evaluating complex functions.

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