Bartle The Elements of Integration exercise 6S
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Let $f_n$ $in$ $L_p(X,chi,mu)$, $1 leq p <+infty$ and let $beta_n$ be defined for $E in chi$ by $$beta_n(E) = left(int_{E} |f_n|^p dmuright)^{1/p}$$ and suposse that $(f_n)$ is a Cauchy sequence in $L_p$. If $epsilon >0$, then exists a $delta(epsilon)>0$ such that if $E in chi$ and $mu(E)<delta(epsilon)$, then $beta_n(E)<epsilon$ for all $n in mathbb{N}$.
I'm trying to solve the problem above, I already know that the limit of $beta_n(E)$ exists for every $E in chi$ since $(f_n)$ is a Cauchy sequence. My attempt to solve the problem is to define the set $$B={ E in chi ; beta_n(E)<epsilon
,,,forall n in mathbb{N},,, and,,, mu(E)<+infty }.$$ The set $B$ is clearly not empty and if $mu(E) = 0$ for all $E in B$ the proposition is proved in that case. The other case consists of the existence of one $M in B$ such that $0<mu(M)$, here is the part that I'm stuck and don't know how to proceed to prove the existence of such $delta$. Every hint in how to use the hypothesis that $(f_n)$ is a Cauchy sequence to solve the second case will be much appreciated, thank you.
real-analysis lebesgue-integral lp-spaces
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
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add a comment |
$begingroup$
Let $f_n$ $in$ $L_p(X,chi,mu)$, $1 leq p <+infty$ and let $beta_n$ be defined for $E in chi$ by $$beta_n(E) = left(int_{E} |f_n|^p dmuright)^{1/p}$$ and suposse that $(f_n)$ is a Cauchy sequence in $L_p$. If $epsilon >0$, then exists a $delta(epsilon)>0$ such that if $E in chi$ and $mu(E)<delta(epsilon)$, then $beta_n(E)<epsilon$ for all $n in mathbb{N}$.
I'm trying to solve the problem above, I already know that the limit of $beta_n(E)$ exists for every $E in chi$ since $(f_n)$ is a Cauchy sequence. My attempt to solve the problem is to define the set $$B={ E in chi ; beta_n(E)<epsilon
,,,forall n in mathbb{N},,, and,,, mu(E)<+infty }.$$ The set $B$ is clearly not empty and if $mu(E) = 0$ for all $E in B$ the proposition is proved in that case. The other case consists of the existence of one $M in B$ such that $0<mu(M)$, here is the part that I'm stuck and don't know how to proceed to prove the existence of such $delta$. Every hint in how to use the hypothesis that $(f_n)$ is a Cauchy sequence to solve the second case will be much appreciated, thank you.
real-analysis lebesgue-integral lp-spaces
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
$endgroup$
add a comment |
$begingroup$
Let $f_n$ $in$ $L_p(X,chi,mu)$, $1 leq p <+infty$ and let $beta_n$ be defined for $E in chi$ by $$beta_n(E) = left(int_{E} |f_n|^p dmuright)^{1/p}$$ and suposse that $(f_n)$ is a Cauchy sequence in $L_p$. If $epsilon >0$, then exists a $delta(epsilon)>0$ such that if $E in chi$ and $mu(E)<delta(epsilon)$, then $beta_n(E)<epsilon$ for all $n in mathbb{N}$.
I'm trying to solve the problem above, I already know that the limit of $beta_n(E)$ exists for every $E in chi$ since $(f_n)$ is a Cauchy sequence. My attempt to solve the problem is to define the set $$B={ E in chi ; beta_n(E)<epsilon
,,,forall n in mathbb{N},,, and,,, mu(E)<+infty }.$$ The set $B$ is clearly not empty and if $mu(E) = 0$ for all $E in B$ the proposition is proved in that case. The other case consists of the existence of one $M in B$ such that $0<mu(M)$, here is the part that I'm stuck and don't know how to proceed to prove the existence of such $delta$. Every hint in how to use the hypothesis that $(f_n)$ is a Cauchy sequence to solve the second case will be much appreciated, thank you.
real-analysis lebesgue-integral lp-spaces
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
$endgroup$
Let $f_n$ $in$ $L_p(X,chi,mu)$, $1 leq p <+infty$ and let $beta_n$ be defined for $E in chi$ by $$beta_n(E) = left(int_{E} |f_n|^p dmuright)^{1/p}$$ and suposse that $(f_n)$ is a Cauchy sequence in $L_p$. If $epsilon >0$, then exists a $delta(epsilon)>0$ such that if $E in chi$ and $mu(E)<delta(epsilon)$, then $beta_n(E)<epsilon$ for all $n in mathbb{N}$.
I'm trying to solve the problem above, I already know that the limit of $beta_n(E)$ exists for every $E in chi$ since $(f_n)$ is a Cauchy sequence. My attempt to solve the problem is to define the set $$B={ E in chi ; beta_n(E)<epsilon
,,,forall n in mathbb{N},,, and,,, mu(E)<+infty }.$$ The set $B$ is clearly not empty and if $mu(E) = 0$ for all $E in B$ the proposition is proved in that case. The other case consists of the existence of one $M in B$ such that $0<mu(M)$, here is the part that I'm stuck and don't know how to proceed to prove the existence of such $delta$. Every hint in how to use the hypothesis that $(f_n)$ is a Cauchy sequence to solve the second case will be much appreciated, thank you.
real-analysis lebesgue-integral lp-spaces
real-analysis lebesgue-integral lp-spaces
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
asked Jan 20 at 8:49
Victor RafaelVictor Rafael
454
454
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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Here are some ideas. Observe that
$$
leftlvert beta_n(E)-beta_m(E)rightrvert=leftlvert leftlVert f_nmathbf 1(E)rightrVert_p-leftlVert f_mmathbf 1(E)rightrVert_prightrvert
leqslant leftlVert (f_n-f_n)mathbf 1(E)rightrVert_pleqslant leftlVert f_n-f_mrightrVert_p
$$
hence for each positive $varepsilon$, one can find $n_0$ such that for all $ngeqslant n_0$ and all $Einchi$,
$$
beta_n(E)ltvarepsilon/2+beta_{n_0}(E).
$$
The problem thus reduces to shows the existence of a $delta(varepsilon)$ such that
for all $Einchi$ such that $mu(E)ltdelta(varepsilon)$,
$$
max_{1leqslant jleqslant n_0}beta_j(E)<varepsilon/2.
$$
It suffices to do it for a single function because if $delta_j$ does the job for $f_j$, then take $delta=min_{1leqslant jleqslant n_0}delta_j$.
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
$endgroup$
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
add a comment |
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1 Answer
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active
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votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here are some ideas. Observe that
$$
leftlvert beta_n(E)-beta_m(E)rightrvert=leftlvert leftlVert f_nmathbf 1(E)rightrVert_p-leftlVert f_mmathbf 1(E)rightrVert_prightrvert
leqslant leftlVert (f_n-f_n)mathbf 1(E)rightrVert_pleqslant leftlVert f_n-f_mrightrVert_p
$$
hence for each positive $varepsilon$, one can find $n_0$ such that for all $ngeqslant n_0$ and all $Einchi$,
$$
beta_n(E)ltvarepsilon/2+beta_{n_0}(E).
$$
The problem thus reduces to shows the existence of a $delta(varepsilon)$ such that
for all $Einchi$ such that $mu(E)ltdelta(varepsilon)$,
$$
max_{1leqslant jleqslant n_0}beta_j(E)<varepsilon/2.
$$
It suffices to do it for a single function because if $delta_j$ does the job for $f_j$, then take $delta=min_{1leqslant jleqslant n_0}delta_j$.
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
$endgroup$
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
add a comment |
$begingroup$
Here are some ideas. Observe that
$$
leftlvert beta_n(E)-beta_m(E)rightrvert=leftlvert leftlVert f_nmathbf 1(E)rightrVert_p-leftlVert f_mmathbf 1(E)rightrVert_prightrvert
leqslant leftlVert (f_n-f_n)mathbf 1(E)rightrVert_pleqslant leftlVert f_n-f_mrightrVert_p
$$
hence for each positive $varepsilon$, one can find $n_0$ such that for all $ngeqslant n_0$ and all $Einchi$,
$$
beta_n(E)ltvarepsilon/2+beta_{n_0}(E).
$$
The problem thus reduces to shows the existence of a $delta(varepsilon)$ such that
for all $Einchi$ such that $mu(E)ltdelta(varepsilon)$,
$$
max_{1leqslant jleqslant n_0}beta_j(E)<varepsilon/2.
$$
It suffices to do it for a single function because if $delta_j$ does the job for $f_j$, then take $delta=min_{1leqslant jleqslant n_0}delta_j$.
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
$endgroup$
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
add a comment |
$begingroup$
Here are some ideas. Observe that
$$
leftlvert beta_n(E)-beta_m(E)rightrvert=leftlvert leftlVert f_nmathbf 1(E)rightrVert_p-leftlVert f_mmathbf 1(E)rightrVert_prightrvert
leqslant leftlVert (f_n-f_n)mathbf 1(E)rightrVert_pleqslant leftlVert f_n-f_mrightrVert_p
$$
hence for each positive $varepsilon$, one can find $n_0$ such that for all $ngeqslant n_0$ and all $Einchi$,
$$
beta_n(E)ltvarepsilon/2+beta_{n_0}(E).
$$
The problem thus reduces to shows the existence of a $delta(varepsilon)$ such that
for all $Einchi$ such that $mu(E)ltdelta(varepsilon)$,
$$
max_{1leqslant jleqslant n_0}beta_j(E)<varepsilon/2.
$$
It suffices to do it for a single function because if $delta_j$ does the job for $f_j$, then take $delta=min_{1leqslant jleqslant n_0}delta_j$.
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
$endgroup$
Here are some ideas. Observe that
$$
leftlvert beta_n(E)-beta_m(E)rightrvert=leftlvert leftlVert f_nmathbf 1(E)rightrVert_p-leftlVert f_mmathbf 1(E)rightrVert_prightrvert
leqslant leftlVert (f_n-f_n)mathbf 1(E)rightrVert_pleqslant leftlVert f_n-f_mrightrVert_p
$$
hence for each positive $varepsilon$, one can find $n_0$ such that for all $ngeqslant n_0$ and all $Einchi$,
$$
beta_n(E)ltvarepsilon/2+beta_{n_0}(E).
$$
The problem thus reduces to shows the existence of a $delta(varepsilon)$ such that
for all $Einchi$ such that $mu(E)ltdelta(varepsilon)$,
$$
max_{1leqslant jleqslant n_0}beta_j(E)<varepsilon/2.
$$
It suffices to do it for a single function because if $delta_j$ does the job for $f_j$, then take $delta=min_{1leqslant jleqslant n_0}delta_j$.
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
answered Jan 20 at 10:56
Davide GiraudoDavide Giraudo
127k16151263
127k16151263
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
add a comment |
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
1
1
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
$begingroup$
Your strategy worked just fine! Thank you so much for the help, I found the delta using a equivalent definition of absolutely continuous measures found here math.stackexchange.com/questions/535185/…
$endgroup$
– Victor Rafael
Jan 21 at 3:59
add a comment |
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