Mixing
Fatou's lemma applicability in this case?
$begingroup$
Let $varphi: mathbb{N} to mathbb{R}$; let $g: mathbb{R}^{2} to mathbb{R}$;
let $f_{n}: t mapsto g(varphi (n), t)$ on $mathbb{R}$ for all $n in mathbb{N}$.
Then under suitable conditions we have
$$
int liminf_{n to infty}f_{n} leq liminf_{n to infty}int f_{n}
$$
by Fatou's lemma.
I would like to ask if today the liminf is taken instead with respect to the argument $varphi (n)$, what is a conclusion we can draw similar to the integral inequality? Specifically, do we still have
$int g(liminf_{n to infty}varphi (n), cdot) leq liminf_{n to infty} int g(varphi_{n}, cdot)$ to a certain extent?
real-analysis limits measure-theory convergence lebesgue-integral
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
$endgroup$
add a comment |
$begingroup$
Let $varphi: mathbb{N} to mathbb{R}$; let $g: mathbb{R}^{2} to mathbb{R}$;
let $f_{n}: t mapsto g(varphi (n), t)$ on $mathbb{R}$ for all $n in mathbb{N}$.
Then under suitable conditions we have
$$
int liminf_{n to infty}f_{n} leq liminf_{n to infty}int f_{n}
$$
by Fatou's lemma.
I would like to ask if today the liminf is taken instead with respect to the argument $varphi (n)$, what is a conclusion we can draw similar to the integral inequality? Specifically, do we still have
$int g(liminf_{n to infty}varphi (n), cdot) leq liminf_{n to infty} int g(varphi_{n}, cdot)$ to a certain extent?
real-analysis limits measure-theory convergence lebesgue-integral
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
$endgroup$
add a comment |
$begingroup$
Let $varphi: mathbb{N} to mathbb{R}$; let $g: mathbb{R}^{2} to mathbb{R}$;
let $f_{n}: t mapsto g(varphi (n), t)$ on $mathbb{R}$ for all $n in mathbb{N}$.
Then under suitable conditions we have
$$
int liminf_{n to infty}f_{n} leq liminf_{n to infty}int f_{n}
$$
by Fatou's lemma.
I would like to ask if today the liminf is taken instead with respect to the argument $varphi (n)$, what is a conclusion we can draw similar to the integral inequality? Specifically, do we still have
$int g(liminf_{n to infty}varphi (n), cdot) leq liminf_{n to infty} int g(varphi_{n}, cdot)$ to a certain extent?
real-analysis limits measure-theory convergence lebesgue-integral
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
$endgroup$
Let $varphi: mathbb{N} to mathbb{R}$; let $g: mathbb{R}^{2} to mathbb{R}$;
let $f_{n}: t mapsto g(varphi (n), t)$ on $mathbb{R}$ for all $n in mathbb{N}$.
Then under suitable conditions we have
$$
int liminf_{n to infty}f_{n} leq liminf_{n to infty}int f_{n}
$$
by Fatou's lemma.
I would like to ask if today the liminf is taken instead with respect to the argument $varphi (n)$, what is a conclusion we can draw similar to the integral inequality? Specifically, do we still have
$int g(liminf_{n to infty}varphi (n), cdot) leq liminf_{n to infty} int g(varphi_{n}, cdot)$ to a certain extent?
real-analysis limits measure-theory convergence lebesgue-integral
real-analysis limits measure-theory convergence lebesgue-integral
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
asked Jan 2 at 18:54
Gary MooreGary Moore
17.2k21545
17.2k21545
add a comment |
add a comment |
1 Answer
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$begingroup$
Without continuity of $g$, we have no reason to think $g(liminf_{n to infty }varphi(n), cdot)$ is related in any way to $g(varphi(n), cdot)$.
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
$endgroup$
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Without continuity of $g$, we have no reason to think $g(liminf_{n to infty }varphi(n), cdot)$ is related in any way to $g(varphi(n), cdot)$.
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
$endgroup$
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
add a comment |
$begingroup$
Without continuity of $g$, we have no reason to think $g(liminf_{n to infty }varphi(n), cdot)$ is related in any way to $g(varphi(n), cdot)$.
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
$endgroup$
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
add a comment |
$begingroup$
Without continuity of $g$, we have no reason to think $g(liminf_{n to infty }varphi(n), cdot)$ is related in any way to $g(varphi(n), cdot)$.
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
$endgroup$
Without continuity of $g$, we have no reason to think $g(liminf_{n to infty }varphi(n), cdot)$ is related in any way to $g(varphi(n), cdot)$.
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
answered Jan 2 at 19:02
Robert IsraelRobert Israel
319k23209459
319k23209459
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
add a comment |
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
$begingroup$
Hello, I guess I deliberately imposed no assumptions on any functions involved to potentially allow the most general result related. You are welcome to provide any information; thanks.
$endgroup$
– Gary Moore
Jan 2 at 19:10
add a comment |
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