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Bounding the integral over a set by square root of its measure
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While studying for an upcoming qualifying exam, I have been attempting to solve questions from previous exams. One question has been bothering me for some time now.
Let $f:mathbb Rtomathbb R$ be a measurable function such that for some $C>0$
$$m(f(x))leq Clambda^-2, textfor all $lambda>0$.$$
Prove that there is some $C'>0$ such that
$$int_E|f(x)| dxleq C'sqrtm(E), textfor all measurable $Esubsetmathbb R$.$$
If I can show that $fin L^2(mathbb R)$, then a relatively easy argument involving Jensen's inequality can establish the desired inequality with $C'=|f|_2$. But I am unable to show that $fin L^2(mathbb R)$ (and I'm not very confident that the hypotheses prove this), and have not found a good alternative approach.
Any advice as to how I should proceed would be greatly appreciated.
real-analysis measure-theory lebesgue-measure
asked Aug 3 at 23:21
Aweygan
11.8k21436
add a comment |Â
up vote
1
down vote
favorite
While studying for an upcoming qualifying exam, I have been attempting to solve questions from previous exams. One question has been bothering me for some time now.
Let $f:mathbb Rtomathbb R$ be a measurable function such that for some $C>0$
$$m(f(x))leq Clambda^-2, textfor all $lambda>0$.$$
Prove that there is some $C'>0$ such that
$$int_E|f(x)| dxleq C'sqrtm(E), textfor all measurable $Esubsetmathbb R$.$$
If I can show that $fin L^2(mathbb R)$, then a relatively easy argument involving Jensen's inequality can establish the desired inequality with $C'=|f|_2$. But I am unable to show that $fin L^2(mathbb R)$ (and I'm not very confident that the hypotheses prove this), and have not found a good alternative approach.
Any advice as to how I should proceed would be greatly appreciated.
real-analysis measure-theory lebesgue-measure
asked Aug 3 at 23:21
Aweygan
11.8k21436
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
While studying for an upcoming qualifying exam, I have been attempting to solve questions from previous exams. One question has been bothering me for some time now.
Let $f:mathbb Rtomathbb R$ be a measurable function such that for some $C>0$
$$m(f(x))leq Clambda^-2, textfor all $lambda>0$.$$
Prove that there is some $C'>0$ such that
$$int_E|f(x)| dxleq C'sqrtm(E), textfor all measurable $Esubsetmathbb R$.$$
If I can show that $fin L^2(mathbb R)$, then a relatively easy argument involving Jensen's inequality can establish the desired inequality with $C'=|f|_2$. But I am unable to show that $fin L^2(mathbb R)$ (and I'm not very confident that the hypotheses prove this), and have not found a good alternative approach.
Any advice as to how I should proceed would be greatly appreciated.
real-analysis measure-theory lebesgue-measure
asked Aug 3 at 23:21
Aweygan
11.8k21436
While studying for an upcoming qualifying exam, I have been attempting to solve questions from previous exams. One question has been bothering me for some time now.
Let $f:mathbb Rtomathbb R$ be a measurable function such that for some $C>0$
$$m(f(x))leq Clambda^-2, textfor all $lambda>0$.$$
Prove that there is some $C'>0$ such that
$$int_E|f(x)| dxleq C'sqrtm(E), textfor all measurable $Esubsetmathbb R$.$$
If I can show that $fin L^2(mathbb R)$, then a relatively easy argument involving Jensen's inequality can establish the desired inequality with $C'=|f|_2$. But I am unable to show that $fin L^2(mathbb R)$ (and I'm not very confident that the hypotheses prove this), and have not found a good alternative approach.
Any advice as to how I should proceed would be greatly appreciated.
real-analysis measure-theory lebesgue-measure
asked Aug 3 at 23:21
Aweygan
11.8k21436
edited Aug 3 at 23:56
asked Aug 3 at 23:21
Aweygan
11.8k21436
asked Aug 3 at 23:21
Aweygan
11.8k21436
asked Aug 3 at 23:21
Aweygan
11.8k21436
11.8k21436
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
Here's a somewhat detailed outline of a proof.
Recall the following formula for the integral of a function:
$$
int_E |f(x)|, dx = int_0^infty mf(x), dlambda.
$$
Now split the last integral into two integrals $int_0^t$ and $int_t^infty$ for $t$ to be determined. Bound the first by $tm(E)$ and the second by $C/t$ and then find the $t$ that minimizes the upper bound.
To see that your hypotheses do not imply $fin L^2(mathbbR)$ (or even $fin L^2_loc(mathbbR)$) consider $f(x)=|x|^-1/2$.
answered 2 days ago
Jose27
6,1951912
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Here's a somewhat detailed outline of a proof.
Recall the following formula for the integral of a function:
$$
int_E |f(x)|, dx = int_0^infty mf(x), dlambda.
$$
Now split the last integral into two integrals $int_0^t$ and $int_t^infty$ for $t$ to be determined. Bound the first by $tm(E)$ and the second by $C/t$ and then find the $t$ that minimizes the upper bound.
To see that your hypotheses do not imply $fin L^2(mathbbR)$ (or even $fin L^2_loc(mathbbR)$) consider $f(x)=|x|^-1/2$.
answered 2 days ago
Jose27
6,1951912
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
add a comment |Â
up vote
4
down vote
accepted
Here's a somewhat detailed outline of a proof.
Recall the following formula for the integral of a function:
$$
int_E |f(x)|, dx = int_0^infty mf(x), dlambda.
$$
Now split the last integral into two integrals $int_0^t$ and $int_t^infty$ for $t$ to be determined. Bound the first by $tm(E)$ and the second by $C/t$ and then find the $t$ that minimizes the upper bound.
To see that your hypotheses do not imply $fin L^2(mathbbR)$ (or even $fin L^2_loc(mathbbR)$) consider $f(x)=|x|^-1/2$.
answered 2 days ago
Jose27
6,1951912
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Here's a somewhat detailed outline of a proof.
Recall the following formula for the integral of a function:
$$
int_E |f(x)|, dx = int_0^infty mf(x), dlambda.
$$
Now split the last integral into two integrals $int_0^t$ and $int_t^infty$ for $t$ to be determined. Bound the first by $tm(E)$ and the second by $C/t$ and then find the $t$ that minimizes the upper bound.
To see that your hypotheses do not imply $fin L^2(mathbbR)$ (or even $fin L^2_loc(mathbbR)$) consider $f(x)=|x|^-1/2$.
answered 2 days ago
Jose27
6,1951912
Here's a somewhat detailed outline of a proof.
Recall the following formula for the integral of a function:
$$
int_E |f(x)|, dx = int_0^infty mf(x), dlambda.
$$
Now split the last integral into two integrals $int_0^t$ and $int_t^infty$ for $t$ to be determined. Bound the first by $tm(E)$ and the second by $C/t$ and then find the $t$ that minimizes the upper bound.
To see that your hypotheses do not imply $fin L^2(mathbbR)$ (or even $fin L^2_loc(mathbbR)$) consider $f(x)=|x|^-1/2$.
answered 2 days ago
Jose27
6,1951912
edited 2 days ago
answered 2 days ago
Jose27
6,1951912
answered 2 days ago
Jose27
6,1951912
answered 2 days ago
Jose27
6,1951912
6,1951912
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
add a comment |Â
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
Very nice answer!
– amsmath
2 days ago
Very nice answer!
– amsmath
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
Ahh thank you. I was not familiar with that integral formula.
– Aweygan
2 days ago
add a comment |Â
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