A sequence of test functions that converges to a charscteristic function from beiow.
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Let $E$ be a Borel bounded set of $mathbb{R}^{n}$ and $chi$ be the characteristic function of $E$. How would you construct a sequence $ chi_{n}$ of nonnegative test functions bounded above by $chi$ that tends to $chi$, as $ntoinfty$?
real-analysis distribution-theory
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
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add a comment |
$begingroup$
Let $E$ be a Borel bounded set of $mathbb{R}^{n}$ and $chi$ be the characteristic function of $E$. How would you construct a sequence $ chi_{n}$ of nonnegative test functions bounded above by $chi$ that tends to $chi$, as $ntoinfty$?
real-analysis distribution-theory
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
$endgroup$
$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16
add a comment |
$begingroup$
Let $E$ be a Borel bounded set of $mathbb{R}^{n}$ and $chi$ be the characteristic function of $E$. How would you construct a sequence $ chi_{n}$ of nonnegative test functions bounded above by $chi$ that tends to $chi$, as $ntoinfty$?
real-analysis distribution-theory
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
$endgroup$
Let $E$ be a Borel bounded set of $mathbb{R}^{n}$ and $chi$ be the characteristic function of $E$. How would you construct a sequence $ chi_{n}$ of nonnegative test functions bounded above by $chi$ that tends to $chi$, as $ntoinfty$?
real-analysis distribution-theory
real-analysis distribution-theory
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
edited Dec 9 '18 at 2:17
M. Rahmat
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
asked Dec 8 '18 at 22:11
M. RahmatM. Rahmat
291212
291212
$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16
add a comment |
$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16
$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16
$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16
add a comment |
1 Answer
1
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Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0leq f_n leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_nequiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
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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
votes
active
oldest
votes
$begingroup$
Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0leq f_n leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_nequiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
$endgroup$
add a comment |
$begingroup$
Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0leq f_n leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_nequiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
$endgroup$
add a comment |
$begingroup$
Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0leq f_n leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_nequiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
$endgroup$
Let $E$ be the set of all irrational numbers in $(0,1)$. If $f_n$'s are test functions such that $0leq f_n leq I_E$ then $f_n(x)=0$ for all rational numbers $x$ in $(0,1)$. Since test functions are continuous this implies $f_nequiv 0$ on $(0,1)$. I no sense does $(f_n)$ converge to $I_E$.
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
answered Dec 8 '18 at 23:55
Kavi Rama MurthyKavi Rama Murthy
60.3k42161
60.3k42161
add a comment |
add a comment |
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$begingroup$
tends to $chi$ how? (Also, please typeset Rn properly)
$endgroup$
– zhw.
Dec 8 '18 at 22:16