Mixing
Defining an “improper” Lebesgue integral?
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I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $frac{sin x}{x}$. While I understand perfectly well why such a function is not Lebesgue integrable, I can't help but wonder if there is some meaningful way to extend the idea of the Lebesgue integral, perhaps as an "improper" Lebesgue integral.
The first idea that comes to mind is looking at all measurable functions which have bounded $L^p$ norms over all bounded intervals:
$$mathcal{L}^p(mathbb{R}) := left{, f : sup_{-infty < a < b < infty} left|left| ,chi_{(a,b)} ,f ,right|right|_{L^P} < infty
right},$$
where $chi_{(a,b)}$ is the characteristic function of the interval $(a,b)$.
I realize that we are being a bit arbitrary in the requirement that the intervals be bounded, but at the same time we aren't restricting ourselves to something as rigid as symmetric intervals (like a principal value integral might do). Would such a space have any meaningful properties, or would all of the important theorems from measure theory and functional analysis fall apart? And is there any nice way of classifying the functions that live in the space $mathcal{L}^p(mathbb{R})setminus L^p(mathbb{R})$?
measure-theory lebesgue-measure lp-spaces
asked Jan 27 at 2:57
PatchPatch
1,9311229
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add a comment |
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I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $frac{sin x}{x}$. While I understand perfectly well why such a function is not Lebesgue integrable, I can't help but wonder if there is some meaningful way to extend the idea of the Lebesgue integral, perhaps as an "improper" Lebesgue integral.
The first idea that comes to mind is looking at all measurable functions which have bounded $L^p$ norms over all bounded intervals:
$$mathcal{L}^p(mathbb{R}) := left{, f : sup_{-infty < a < b < infty} left|left| ,chi_{(a,b)} ,f ,right|right|_{L^P} < infty
right},$$
where $chi_{(a,b)}$ is the characteristic function of the interval $(a,b)$.
I realize that we are being a bit arbitrary in the requirement that the intervals be bounded, but at the same time we aren't restricting ourselves to something as rigid as symmetric intervals (like a principal value integral might do). Would such a space have any meaningful properties, or would all of the important theorems from measure theory and functional analysis fall apart? And is there any nice way of classifying the functions that live in the space $mathcal{L}^p(mathbb{R})setminus L^p(mathbb{R})$?
measure-theory lebesgue-measure lp-spaces
asked Jan 27 at 2:57
PatchPatch
1,9311229
$endgroup$
$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58
add a comment |
$begingroup$
I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $frac{sin x}{x}$. While I understand perfectly well why such a function is not Lebesgue integrable, I can't help but wonder if there is some meaningful way to extend the idea of the Lebesgue integral, perhaps as an "improper" Lebesgue integral.
The first idea that comes to mind is looking at all measurable functions which have bounded $L^p$ norms over all bounded intervals:
$$mathcal{L}^p(mathbb{R}) := left{, f : sup_{-infty < a < b < infty} left|left| ,chi_{(a,b)} ,f ,right|right|_{L^P} < infty
right},$$
where $chi_{(a,b)}$ is the characteristic function of the interval $(a,b)$.
I realize that we are being a bit arbitrary in the requirement that the intervals be bounded, but at the same time we aren't restricting ourselves to something as rigid as symmetric intervals (like a principal value integral might do). Would such a space have any meaningful properties, or would all of the important theorems from measure theory and functional analysis fall apart? And is there any nice way of classifying the functions that live in the space $mathcal{L}^p(mathbb{R})setminus L^p(mathbb{R})$?
measure-theory lebesgue-measure lp-spaces
asked Jan 27 at 2:57
PatchPatch
1,9311229
$endgroup$
I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $frac{sin x}{x}$. While I understand perfectly well why such a function is not Lebesgue integrable, I can't help but wonder if there is some meaningful way to extend the idea of the Lebesgue integral, perhaps as an "improper" Lebesgue integral.
The first idea that comes to mind is looking at all measurable functions which have bounded $L^p$ norms over all bounded intervals:
$$mathcal{L}^p(mathbb{R}) := left{, f : sup_{-infty < a < b < infty} left|left| ,chi_{(a,b)} ,f ,right|right|_{L^P} < infty
right},$$
where $chi_{(a,b)}$ is the characteristic function of the interval $(a,b)$.
I realize that we are being a bit arbitrary in the requirement that the intervals be bounded, but at the same time we aren't restricting ourselves to something as rigid as symmetric intervals (like a principal value integral might do). Would such a space have any meaningful properties, or would all of the important theorems from measure theory and functional analysis fall apart? And is there any nice way of classifying the functions that live in the space $mathcal{L}^p(mathbb{R})setminus L^p(mathbb{R})$?
measure-theory lebesgue-measure lp-spaces
measure-theory lebesgue-measure lp-spaces
asked Jan 27 at 2:57
PatchPatch
1,9311229
asked Jan 27 at 2:57
PatchPatch
1,9311229
asked Jan 27 at 2:57
PatchPatch
1,9311229
asked Jan 27 at 2:57
PatchPatch
1,9311229
asked Jan 27 at 2:57
PatchPatch
1,9311229
1,9311229
$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58
add a comment |
$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58
$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58
add a comment |
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$begingroup$
Possible duplicate of Is there a general theory of the "improper" Lebesgue integral?
$endgroup$
– JavaMan
Jan 27 at 3:00
$begingroup$
I saw that post, but I'm considering a particular kind of integral space. I'm not quite looking for a fully-generalized version of the Lebesgue integral the way I understood the other poster to be looking for.
$endgroup$
– Patch
Jan 27 at 3:04
$begingroup$
That's a fair point. Your question is different enough that I've retracted my close vote.
$endgroup$
– JavaMan
Jan 27 at 3:50
$begingroup$
Your $mathcal L^p(mathbb R)$ is just the same as $L^p(mathbb R)$. Functions such as $sin(x)/x$ are improperly Riemann integrable, but their absolute values are not.
$endgroup$
– Robert Israel
Jan 27 at 3:58