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Extend convergence on dense subset to the space
The 2019 Stack Overflow Developer Survey Results Are InHow do I look up applications of this type of convergence?Dominated convergence theorem for $limint_{0}^{n}(1-x/n)^{n}x^sdx$, $s<-1$.Generalisation of Dominated Convergence TheoremMeasure Convergence Version of Lebesgue Dominated Convergence TheoremTaking limit inside integrationWhy is the Monotone Convergence Theorem restricted to a nonnegative function sequence?Is this a special case of the bounded convergence theorem?Dominated convergence theorem of Lebesgue measurable sets using monotone convergence theoremA density proof of the Dominated Convergence theoremHow to find the value for $beta$
$begingroup$
The motivation is to see when I can use the Dominated Convergence Theorem or the Monotone Convergence Theorem when all I know is the convergence of the function pointwise in a dense subset.
Let $(X,mathcal{B}(X))$ be a measurable space with the Borel sigma algebra. Let $Dsubset X$ be a dense subset of X. Let $f_n$ and $f$ be measurable functions such that, for every $din D$, $f_n(d)to f(d)$. Let $mu$ be any arbitrary measure supported by the space. I´m interested in conditions such that
$$lim_{ntoinfty}int_Xf_ndmu=int_Xfdmu$$
One I have in mind is when $f$ and $f_n$ are continuous, so as the set $D$ is dense, I can always construct a sequence within $D$ tending to any point $xin Xsetminus D$ and pass through the limit to x. I'm tempted to say the argument follows when the set of discontinuities of $f_n$ and $f$ is of measure zero too.
Are any other conditions someone recall to extend the pointwise convergence from the dense subset to the entire set, or to make work the DCT or MTC when we only know the pointwise convergence in the dense subset?
Thanks!
real-analysis probability convergence
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
$endgroup$
add a comment |
$begingroup$
The motivation is to see when I can use the Dominated Convergence Theorem or the Monotone Convergence Theorem when all I know is the convergence of the function pointwise in a dense subset.
Let $(X,mathcal{B}(X))$ be a measurable space with the Borel sigma algebra. Let $Dsubset X$ be a dense subset of X. Let $f_n$ and $f$ be measurable functions such that, for every $din D$, $f_n(d)to f(d)$. Let $mu$ be any arbitrary measure supported by the space. I´m interested in conditions such that
$$lim_{ntoinfty}int_Xf_ndmu=int_Xfdmu$$
One I have in mind is when $f$ and $f_n$ are continuous, so as the set $D$ is dense, I can always construct a sequence within $D$ tending to any point $xin Xsetminus D$ and pass through the limit to x. I'm tempted to say the argument follows when the set of discontinuities of $f_n$ and $f$ is of measure zero too.
Are any other conditions someone recall to extend the pointwise convergence from the dense subset to the entire set, or to make work the DCT or MTC when we only know the pointwise convergence in the dense subset?
Thanks!
real-analysis probability convergence
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
$endgroup$
add a comment |
$begingroup$
The motivation is to see when I can use the Dominated Convergence Theorem or the Monotone Convergence Theorem when all I know is the convergence of the function pointwise in a dense subset.
Let $(X,mathcal{B}(X))$ be a measurable space with the Borel sigma algebra. Let $Dsubset X$ be a dense subset of X. Let $f_n$ and $f$ be measurable functions such that, for every $din D$, $f_n(d)to f(d)$. Let $mu$ be any arbitrary measure supported by the space. I´m interested in conditions such that
$$lim_{ntoinfty}int_Xf_ndmu=int_Xfdmu$$
One I have in mind is when $f$ and $f_n$ are continuous, so as the set $D$ is dense, I can always construct a sequence within $D$ tending to any point $xin Xsetminus D$ and pass through the limit to x. I'm tempted to say the argument follows when the set of discontinuities of $f_n$ and $f$ is of measure zero too.
Are any other conditions someone recall to extend the pointwise convergence from the dense subset to the entire set, or to make work the DCT or MTC when we only know the pointwise convergence in the dense subset?
Thanks!
real-analysis probability convergence
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
$endgroup$
The motivation is to see when I can use the Dominated Convergence Theorem or the Monotone Convergence Theorem when all I know is the convergence of the function pointwise in a dense subset.
Let $(X,mathcal{B}(X))$ be a measurable space with the Borel sigma algebra. Let $Dsubset X$ be a dense subset of X. Let $f_n$ and $f$ be measurable functions such that, for every $din D$, $f_n(d)to f(d)$. Let $mu$ be any arbitrary measure supported by the space. I´m interested in conditions such that
$$lim_{ntoinfty}int_Xf_ndmu=int_Xfdmu$$
One I have in mind is when $f$ and $f_n$ are continuous, so as the set $D$ is dense, I can always construct a sequence within $D$ tending to any point $xin Xsetminus D$ and pass through the limit to x. I'm tempted to say the argument follows when the set of discontinuities of $f_n$ and $f$ is of measure zero too.
Are any other conditions someone recall to extend the pointwise convergence from the dense subset to the entire set, or to make work the DCT or MTC when we only know the pointwise convergence in the dense subset?
Thanks!
real-analysis probability convergence
real-analysis probability convergence
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
asked Mar 21 at 15:47
Zeky MurraZeky Murra
1407
1407
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|le h$ for all $nge 1$ with $hin L_1$ and the sequence of functions ${g_n}$ is pointwise equicontinouos $mu$-a.e. Then for $xin A$ with $mu(A^c)=0$ and any $epsilon>0$, there exists $x'in D$ s.t.
$$
limsup_{ntoinfty}|g_n(x)|le limsup_{ntoinfty}|g_n(x)-g_n(x')|+lim_{ntoinfty}|g_n(x')|< epsilon,
$$
which implies that $g_nto 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $xin X$ and every $epsilon>0$, there exists $delta_x>0$ s.t.
$$
limsup_{ntoinfty}|g_n(x)-g_n(y)|<epsilon
$$
whenever $|x-y|<delta_x$.
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
$endgroup$
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
add a comment |
Your Answer
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1 Answer
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$begingroup$
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|le h$ for all $nge 1$ with $hin L_1$ and the sequence of functions ${g_n}$ is pointwise equicontinouos $mu$-a.e. Then for $xin A$ with $mu(A^c)=0$ and any $epsilon>0$, there exists $x'in D$ s.t.
$$
limsup_{ntoinfty}|g_n(x)|le limsup_{ntoinfty}|g_n(x)-g_n(x')|+lim_{ntoinfty}|g_n(x')|< epsilon,
$$
which implies that $g_nto 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $xin X$ and every $epsilon>0$, there exists $delta_x>0$ s.t.
$$
limsup_{ntoinfty}|g_n(x)-g_n(y)|<epsilon
$$
whenever $|x-y|<delta_x$.
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
$endgroup$
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
add a comment |
$begingroup$
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|le h$ for all $nge 1$ with $hin L_1$ and the sequence of functions ${g_n}$ is pointwise equicontinouos $mu$-a.e. Then for $xin A$ with $mu(A^c)=0$ and any $epsilon>0$, there exists $x'in D$ s.t.
$$
limsup_{ntoinfty}|g_n(x)|le limsup_{ntoinfty}|g_n(x)-g_n(x')|+lim_{ntoinfty}|g_n(x')|< epsilon,
$$
which implies that $g_nto 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $xin X$ and every $epsilon>0$, there exists $delta_x>0$ s.t.
$$
limsup_{ntoinfty}|g_n(x)-g_n(y)|<epsilon
$$
whenever $|x-y|<delta_x$.
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
$endgroup$
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
add a comment |
$begingroup$
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|le h$ for all $nge 1$ with $hin L_1$ and the sequence of functions ${g_n}$ is pointwise equicontinouos $mu$-a.e. Then for $xin A$ with $mu(A^c)=0$ and any $epsilon>0$, there exists $x'in D$ s.t.
$$
limsup_{ntoinfty}|g_n(x)|le limsup_{ntoinfty}|g_n(x)-g_n(x')|+lim_{ntoinfty}|g_n(x')|< epsilon,
$$
which implies that $g_nto 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $xin X$ and every $epsilon>0$, there exists $delta_x>0$ s.t.
$$
limsup_{ntoinfty}|g_n(x)-g_n(y)|<epsilon
$$
whenever $|x-y|<delta_x$.
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
$endgroup$
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|le h$ for all $nge 1$ with $hin L_1$ and the sequence of functions ${g_n}$ is pointwise equicontinouos $mu$-a.e. Then for $xin A$ with $mu(A^c)=0$ and any $epsilon>0$, there exists $x'in D$ s.t.
$$
limsup_{ntoinfty}|g_n(x)|le limsup_{ntoinfty}|g_n(x)-g_n(x')|+lim_{ntoinfty}|g_n(x')|< epsilon,
$$
which implies that $g_nto 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $xin X$ and every $epsilon>0$, there exists $delta_x>0$ s.t.
$$
limsup_{ntoinfty}|g_n(x)-g_n(y)|<epsilon
$$
whenever $|x-y|<delta_x$.
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
edited Mar 21 at 18:33
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
answered Mar 21 at 17:31
d.k.o.d.k.o.
10.6k730
10.6k730
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
add a comment |
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
Thanks! this is very informative. Just one more question, is $A$ any set or does it has a particular property?
$endgroup$
– Zeky Murra
Mar 21 at 17:39
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
$begingroup$
We know that ${g_n}$ is not equicontinuous at each $xin A$.
$endgroup$
– d.k.o.
Mar 21 at 18:24
add a comment |
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