Pushforward formula for Lebesgue Stieltjes Measure
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I got to have this problem in my hand.
Problem
Let $F: mathbbRrightarrow mathbbR$ be an increasing, right continuous function, and let $phi :mathbbRrightarrow mathbbR$ be a continuous increasing invertible function. Let $mu_F$ and $mu_Fcirc phi$ be the Lebesgue-Stieljes measure associated to $F$ and $Fcirc phi$ respectively. Show that if $fin L^1(mu_F)$, then $fcirc phi in L^1(mu_Fcirc phi)$ and $$ int f dmu_F = int fcirc phi dmu_Fcirc phi$$ Hint: It is enough to consider non-negative $f$ and to prove the inequality $int fcirc phi dmu_Fcirc phileq int fmu_F$.
There were many situation I have studied this kind of problem and I was thinking of using simple function and generalize it to arbitrary $L^1$ functions. And I think the method just works here too. But I can not understand the hint on this problem. Because I started from simple, the hint for non negative function seems superfluous but makes sense. But I don't know how to use the hint for inequality. I will be happy with any kind of hint or comment on it. Thank you in advance!
lebesgue-integral lebesgue-measure pushforward
edited Aug 16 at 11:46
Ethan Bolker
36k54299
asked Aug 16 at 2:44
Lev Ban
56916
add a comment |Â
up vote
1
down vote
favorite
I got to have this problem in my hand.
Problem
Let $F: mathbbRrightarrow mathbbR$ be an increasing, right continuous function, and let $phi :mathbbRrightarrow mathbbR$ be a continuous increasing invertible function. Let $mu_F$ and $mu_Fcirc phi$ be the Lebesgue-Stieljes measure associated to $F$ and $Fcirc phi$ respectively. Show that if $fin L^1(mu_F)$, then $fcirc phi in L^1(mu_Fcirc phi)$ and $$ int f dmu_F = int fcirc phi dmu_Fcirc phi$$ Hint: It is enough to consider non-negative $f$ and to prove the inequality $int fcirc phi dmu_Fcirc phileq int fmu_F$.
There were many situation I have studied this kind of problem and I was thinking of using simple function and generalize it to arbitrary $L^1$ functions. And I think the method just works here too. But I can not understand the hint on this problem. Because I started from simple, the hint for non negative function seems superfluous but makes sense. But I don't know how to use the hint for inequality. I will be happy with any kind of hint or comment on it. Thank you in advance!
lebesgue-integral lebesgue-measure pushforward
edited Aug 16 at 11:46
Ethan Bolker
36k54299
asked Aug 16 at 2:44
Lev Ban
56916
1
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I got to have this problem in my hand.
Problem
Let $F: mathbbRrightarrow mathbbR$ be an increasing, right continuous function, and let $phi :mathbbRrightarrow mathbbR$ be a continuous increasing invertible function. Let $mu_F$ and $mu_Fcirc phi$ be the Lebesgue-Stieljes measure associated to $F$ and $Fcirc phi$ respectively. Show that if $fin L^1(mu_F)$, then $fcirc phi in L^1(mu_Fcirc phi)$ and $$ int f dmu_F = int fcirc phi dmu_Fcirc phi$$ Hint: It is enough to consider non-negative $f$ and to prove the inequality $int fcirc phi dmu_Fcirc phileq int fmu_F$.
There were many situation I have studied this kind of problem and I was thinking of using simple function and generalize it to arbitrary $L^1$ functions. And I think the method just works here too. But I can not understand the hint on this problem. Because I started from simple, the hint for non negative function seems superfluous but makes sense. But I don't know how to use the hint for inequality. I will be happy with any kind of hint or comment on it. Thank you in advance!
lebesgue-integral lebesgue-measure pushforward
edited Aug 16 at 11:46
Ethan Bolker
36k54299
asked Aug 16 at 2:44
Lev Ban
56916
I got to have this problem in my hand.
Problem
Let $F: mathbbRrightarrow mathbbR$ be an increasing, right continuous function, and let $phi :mathbbRrightarrow mathbbR$ be a continuous increasing invertible function. Let $mu_F$ and $mu_Fcirc phi$ be the Lebesgue-Stieljes measure associated to $F$ and $Fcirc phi$ respectively. Show that if $fin L^1(mu_F)$, then $fcirc phi in L^1(mu_Fcirc phi)$ and $$ int f dmu_F = int fcirc phi dmu_Fcirc phi$$ Hint: It is enough to consider non-negative $f$ and to prove the inequality $int fcirc phi dmu_Fcirc phileq int fmu_F$.
There were many situation I have studied this kind of problem and I was thinking of using simple function and generalize it to arbitrary $L^1$ functions. And I think the method just works here too. But I can not understand the hint on this problem. Because I started from simple, the hint for non negative function seems superfluous but makes sense. But I don't know how to use the hint for inequality. I will be happy with any kind of hint or comment on it. Thank you in advance!
lebesgue-integral lebesgue-measure pushforward
edited Aug 16 at 11:46
Ethan Bolker
36k54299
asked Aug 16 at 2:44
Lev Ban
56916
edited Aug 16 at 11:46
Ethan Bolker
36k54299
edited Aug 16 at 11:46
Ethan Bolker
36k54299
edited Aug 16 at 11:46
Ethan Bolker
36k54299
36k54299
asked Aug 16 at 2:44
Lev Ban
56916
asked Aug 16 at 2:44
Lev Ban
56916
asked Aug 16 at 2:44
Lev Ban
56916
56916
1
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11
add a comment |Â
1
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11
1
1
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11
add a comment |Â
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1
If you prove the hint for positive functions, you show that $fcirc phi in L^1(mu_Fcirc phi)$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions.
– Dosidis
Aug 16 at 3:25
@Dosidis That make sense! Thank you! But I still don't know how to use the inequality....
– Lev Ban
Aug 16 at 18:11