Mixing
a sequence of characteristic functions converge uniformly to a function (proof)
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Attempt: Let $x in X$. Then, $varphi_n(x) = k2^-n$ $x in E_kn$, and $0$ if $x notin E_kn$. But, $lim_n to inftyvarphi_n(x) = 0$ for all $xin X$. Therefore, $varphi_n(x)$ uniformly converge to 0 on $E_kn$ for all $k$.
In addition, $f(x)$ is defined as $k2^-n le f(x) < (k+1)2^-n$. When we take a limit, by squeeze theorem $lim_nto inftyf(x) = 0$ for all $x in X$.
Is this a complete proof? I think both functions converge uniformly to 0, but how can I connect these two?
Thank you in advance.
real-analysis
asked Aug 3 at 6:24
Sihyun Kim
696210
add a comment |Â
up vote
0
down vote
favorite
Attempt: Let $x in X$. Then, $varphi_n(x) = k2^-n$ $x in E_kn$, and $0$ if $x notin E_kn$. But, $lim_n to inftyvarphi_n(x) = 0$ for all $xin X$. Therefore, $varphi_n(x)$ uniformly converge to 0 on $E_kn$ for all $k$.
In addition, $f(x)$ is defined as $k2^-n le f(x) < (k+1)2^-n$. When we take a limit, by squeeze theorem $lim_nto inftyf(x) = 0$ for all $x in X$.
Is this a complete proof? I think both functions converge uniformly to 0, but how can I connect these two?
Thank you in advance.
real-analysis
asked Aug 3 at 6:24
Sihyun Kim
696210
From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
1
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Attempt: Let $x in X$. Then, $varphi_n(x) = k2^-n$ $x in E_kn$, and $0$ if $x notin E_kn$. But, $lim_n to inftyvarphi_n(x) = 0$ for all $xin X$. Therefore, $varphi_n(x)$ uniformly converge to 0 on $E_kn$ for all $k$.
In addition, $f(x)$ is defined as $k2^-n le f(x) < (k+1)2^-n$. When we take a limit, by squeeze theorem $lim_nto inftyf(x) = 0$ for all $x in X$.
Is this a complete proof? I think both functions converge uniformly to 0, but how can I connect these two?
Thank you in advance.
real-analysis
asked Aug 3 at 6:24
Sihyun Kim
696210
Attempt: Let $x in X$. Then, $varphi_n(x) = k2^-n$ $x in E_kn$, and $0$ if $x notin E_kn$. But, $lim_n to inftyvarphi_n(x) = 0$ for all $xin X$. Therefore, $varphi_n(x)$ uniformly converge to 0 on $E_kn$ for all $k$.
In addition, $f(x)$ is defined as $k2^-n le f(x) < (k+1)2^-n$. When we take a limit, by squeeze theorem $lim_nto inftyf(x) = 0$ for all $x in X$.
Is this a complete proof? I think both functions converge uniformly to 0, but how can I connect these two?
Thank you in advance.
real-analysis
asked Aug 3 at 6:24
Sihyun Kim
696210
asked Aug 3 at 6:24
Sihyun Kim
696210
asked Aug 3 at 6:24
Sihyun Kim
696210
asked Aug 3 at 6:24
Sihyun Kim
696210
696210
From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
1
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35
add a comment |Â
From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
1
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35
From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
1
1
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35
add a comment |Â
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From Bartle's "The Elements of Integration"
– Mason
Aug 3 at 6:27
You have made many wrong statements, some even meaningless. If $lim phi_n (x)=0$ for all $x$ there is no hope of proving this theorem. The statement $lim_nto infty f(x)=0$ does not make sense.
– Kavi Rama Murthy
Aug 3 at 6:32
1
Since this is one of the most basic results in measure theory I suggest that you can read the proof in the book mentioned by Mason and tell us which particular steps you don't follow.
– Kavi Rama Murthy
Aug 3 at 6:35