Lp convergence involving powers
$begingroup$
Let $(X,mu)$ be a finite measure space. Suppose $int_Xlvert f_n-frvert^p,dmuto0$ for $pgeq1$. For $qin(0,p]$ does it hold that $$int_Xlvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}},dmuto0quad?$$
measure-theory
asked May 11 '17 at 13:34
user375366user375366
885138
$endgroup$
add a comment |
$begingroup$
Let $(X,mu)$ be a finite measure space. Suppose $int_Xlvert f_n-frvert^p,dmuto0$ for $pgeq1$. For $qin(0,p]$ does it hold that $$int_Xlvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}},dmuto0quad?$$
measure-theory
asked May 11 '17 at 13:34
user375366user375366
885138
$endgroup$
add a comment |
$begingroup$
Let $(X,mu)$ be a finite measure space. Suppose $int_Xlvert f_n-frvert^p,dmuto0$ for $pgeq1$. For $qin(0,p]$ does it hold that $$int_Xlvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}},dmuto0quad?$$
measure-theory
asked May 11 '17 at 13:34
user375366user375366
885138
$endgroup$
Let $(X,mu)$ be a finite measure space. Suppose $int_Xlvert f_n-frvert^p,dmuto0$ for $pgeq1$. For $qin(0,p]$ does it hold that $$int_Xlvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}},dmuto0quad?$$
measure-theory
measure-theory
asked May 11 '17 at 13:34
user375366user375366
885138
asked May 11 '17 at 13:34
user375366user375366
885138
edited May 11 '17 at 13:52
user375366
asked May 11 '17 at 13:34
user375366user375366
885138
asked May 11 '17 at 13:34
user375366user375366
885138
asked May 11 '17 at 13:34
user375366user375366
885138
885138
add a comment |
add a comment |
1 Answer
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Yes. I will assume that we work on a probability space.
From item 6 of this answer, we know that convergence to $0$ in $mathbb L^1$ of a sequence of random variables $left(X_nright)_{ngeqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define
$$
X_n:= lvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}}.
$$
By the assumption, $f_nto f$ in probability hence by item 3 of this answer,
$X_nto 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality
$$
lvertlvert arvert^q-lvert brvert^qrvert^{frac{p}{q}}
leqslant lvertlvert arvert^q+lvert brvert^qrvert^{frac{p}{q}}
leqslant 2^{p/q-1}left(lvertlvert arvert^p+lvertlvert brvert^pright).
$$
In this way, we are reduced to show the uniform integrability of the sequence $left(lvertlvert f_nrvert^pright)_{ngeqslant 1}$, which is a consequence of $int_Xlvert f_n-frvert^p,dmuto0$.
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
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add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Yes. I will assume that we work on a probability space.
From item 6 of this answer, we know that convergence to $0$ in $mathbb L^1$ of a sequence of random variables $left(X_nright)_{ngeqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define
$$
X_n:= lvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}}.
$$
By the assumption, $f_nto f$ in probability hence by item 3 of this answer,
$X_nto 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality
$$
lvertlvert arvert^q-lvert brvert^qrvert^{frac{p}{q}}
leqslant lvertlvert arvert^q+lvert brvert^qrvert^{frac{p}{q}}
leqslant 2^{p/q-1}left(lvertlvert arvert^p+lvertlvert brvert^pright).
$$
In this way, we are reduced to show the uniform integrability of the sequence $left(lvertlvert f_nrvert^pright)_{ngeqslant 1}$, which is a consequence of $int_Xlvert f_n-frvert^p,dmuto0$.
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
add a comment |
$begingroup$
Yes. I will assume that we work on a probability space.
From item 6 of this answer, we know that convergence to $0$ in $mathbb L^1$ of a sequence of random variables $left(X_nright)_{ngeqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define
$$
X_n:= lvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}}.
$$
By the assumption, $f_nto f$ in probability hence by item 3 of this answer,
$X_nto 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality
$$
lvertlvert arvert^q-lvert brvert^qrvert^{frac{p}{q}}
leqslant lvertlvert arvert^q+lvert brvert^qrvert^{frac{p}{q}}
leqslant 2^{p/q-1}left(lvertlvert arvert^p+lvertlvert brvert^pright).
$$
In this way, we are reduced to show the uniform integrability of the sequence $left(lvertlvert f_nrvert^pright)_{ngeqslant 1}$, which is a consequence of $int_Xlvert f_n-frvert^p,dmuto0$.
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
add a comment |
$begingroup$
Yes. I will assume that we work on a probability space.
From item 6 of this answer, we know that convergence to $0$ in $mathbb L^1$ of a sequence of random variables $left(X_nright)_{ngeqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define
$$
X_n:= lvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}}.
$$
By the assumption, $f_nto f$ in probability hence by item 3 of this answer,
$X_nto 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality
$$
lvertlvert arvert^q-lvert brvert^qrvert^{frac{p}{q}}
leqslant lvertlvert arvert^q+lvert brvert^qrvert^{frac{p}{q}}
leqslant 2^{p/q-1}left(lvertlvert arvert^p+lvertlvert brvert^pright).
$$
In this way, we are reduced to show the uniform integrability of the sequence $left(lvertlvert f_nrvert^pright)_{ngeqslant 1}$, which is a consequence of $int_Xlvert f_n-frvert^p,dmuto0$.
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
Yes. I will assume that we work on a probability space.
From item 6 of this answer, we know that convergence to $0$ in $mathbb L^1$ of a sequence of random variables $left(X_nright)_{ngeqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define
$$
X_n:= lvertlvert f_nrvert^q-lvert frvert^qrvert^{frac{p}{q}}.
$$
By the assumption, $f_nto f$ in probability hence by item 3 of this answer,
$X_nto 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality
$$
lvertlvert arvert^q-lvert brvert^qrvert^{frac{p}{q}}
leqslant lvertlvert arvert^q+lvert brvert^qrvert^{frac{p}{q}}
leqslant 2^{p/q-1}left(lvertlvert arvert^p+lvertlvert brvert^pright).
$$
In this way, we are reduced to show the uniform integrability of the sequence $left(lvertlvert f_nrvert^pright)_{ngeqslant 1}$, which is a consequence of $int_Xlvert f_n-frvert^p,dmuto0$.
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
answered Dec 11 '18 at 21:01
Davide GiraudoDavide Giraudo
127k16151264
127k16151264
add a comment |
add a comment |
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