If h is L-integrable and f + g = h, then f and g also are

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Discussion Overview

The discussion revolves around the properties of Lebesgue integrability, specifically whether the integrability of a function h implies the integrability of its summands f and g, given that h = f + g. Participants explore various scenarios and examples to understand the implications of this relationship.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether f and g must be Lebesgue integrable if h is integrable, suggesting that it seems intuitive but lacks a clear proof.
  • Another participant challenges the initial reasoning, proposing to consider h first and then analyze f and g, indicating a shift in perspective might be beneficial.
  • A different participant asserts that f and g do not necessarily need to be Lebesgue integrable, citing a general principle about properties not passing to summands.
  • One participant presents a specific case where f is a non-integrable function and g is defined as -f, implying this could serve as an extreme example.
  • Another participant constructs a function g based on the values of f, questioning whether g is Lebesgue integrable under certain conditions.
  • There is a discussion about the integrability of g, with one participant suggesting that it can be shown to be integrable by inspection, while another seeks clarification on the proof process.
  • Participants explore the use of min and max functions to define g, with some expressing uncertainty about the implications for integrability.
  • One participant concludes that g is integrable by constructing it from the min and max of f, indicating a successful understanding of the problem.

Areas of Agreement / Disagreement

Participants express differing views on whether the integrability of h guarantees the integrability of f and g, with some asserting that it does not hold universally. The discussion remains unresolved regarding the conditions under which f and g can be integrable.

Contextual Notes

Some arguments rely on specific definitions and properties of Lebesgue integrability, and the discussion includes various assumptions about the functions involved, which may not be universally applicable.

Castilla
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A little question about the basics of Lebesgue integration. There is a theorem: if functions f and g are L-integrable then the function f+g is also L-integrable.

This may be dumb, but I wish to know about the reciprocal lemma. Let h be a L-integrable function and f and g be functions such that h(x) = f(x) + g(x). Are f and g L-integrable? I suppose it is truth because I can't see how, being f a not L-integrable function, h could yet be a L-integrable one. But I cant' see the proof. Does someone got a minute to help? Thanks.
 
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Your problem is a common one -- you've locked yourself into one direction of thinking. You are trying to imagine functions f and g (at least one of which is not integrable), which make an integrable h.

Why not think about it in a different direction? Try imagining h first, and then look at f and g.
 
No, of course, f and g need not be lebesgue integrable. This is a general phenomenon.

If poperty P defined on something with addition and scalar multm and if P is additive and respects scalar mult, and if there are things without property P, then property P does not pass to summands.

Proof:
(deleted - as hurkyl implies, you should look for it yourself)

as a hint, always try to add zero in a clever way
 
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How about f any non-L integrable function, g=-f?
 
That would be in some sense the extreme example. And it also embodies another important principal: wlog we can shift to the origin...
 
Let f(x) be a L-integrable function, P be a fixed number and let g(x) such that

g(x) = f(x) when f(x) belong to [-P, P]
g(x) = P when f(x) > P.
g(x) = -P when f(x) < - P.

The fact is that f(x) is a L-integrable function and also we know that any constant function is L-integrable. Can we say that g(x) is L-integrable?
 
You can say that is Lebesgue integrable by inspection, and I imagine everyone would be happy with that, or at least they would have been if you hadn't indicated that you didn't see why it was, in the maths usage of the word, trivial (not the in belittling sense of the word).
 
Can you give me some hint to prove that g is L-integrable?

Note.- f, the L-integrable function, has bounded domain (let say [A,B]).

Remember I am not saying that if x belongs to [-P,P] then g(x) = |f(x)|. In such case it would be obvious that g is L-integrable (because f, and therefore |f|, is L-integrable). I am saying that if f(x) belongs to [-P,P] then g(x) = |f(x)|. The set of x for which f(x) belongs to [-P,P] don't necesarily is an interval. How to prove that g is L-integrable?

Same observation for the other two cases. I am not saying that if x > P then g(x) = P. The premise is not so easy.
 
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Ah, I must have misread your post, sorry.
 
  • #10
So, reloading:

Facts: f(x) is a L-integrable function (over [A, B]), P be a fixed (positive)number and let g(x) such that

g(x) = f(x) when f(x) belong to [-P, P] (not x but f(x)).
g(x) = P when f(x) > P.
g(x) = -P when f(x) < - P.


Can you tell me if this function g is L-integrable? At least a "yes" or "no".

I have tried, unsuccessfully, the theorem that says that if p and q are L-integrable functions, then r = min (p, q) and s _ max (p, q) are also L-integrable.
 
  • #11
Hmm, I honestly don't know 'instantly', but let's think about it a little. Right, OK, I see what to do (about 5 minutes, actually make it ten, my first, and second thoughts were off the mark).I think the answer is 'yes' it is integrable: we're just cutting off f if it gets too big or too small.

using min max is a bit tricky, but I think you can do it.

consider the functions a(x) = max(0,f(x)) and b(x)=min(0,f(x)). they are both measurable and a+b=f cos of the useful properties of zero. (I would insert a smiley if i didn't think they were ridiculous in a grown man). Now, I'll leave the rest to you: remember a and b are both measurable functions in their own right.
 
  • #12
Only a question, Matt: the functions a(x) and b(x) are directly used in your proof of the L-integrability of function g, or we must use some functions builded as a(x) and b(x) but with P instead of 0??
 
  • #13
Forget the last post, I think I got it.
g(x) is equal to min(P, a(x)) + max (-P, b(x)), each of these two is L-integrable, then g is the sum of L-integrable functions, then is L-integrable.
Thank you, Matt !
 

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