A bounded, continuous function on $mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear...
Where can I find a proof of the following fact?
Any bounded, continuous real-valued function on $mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of indicators of Borel sets
calculus real-analysis measure-theory reference-request borel-sets
edited Nov 26 at 23:57
T. Bongers
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
add a comment |
Where can I find a proof of the following fact?
Any bounded, continuous real-valued function on $mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of indicators of Borel sets
calculus real-analysis measure-theory reference-request borel-sets
edited Nov 26 at 23:57
T. Bongers
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
add a comment |
Where can I find a proof of the following fact?
Any bounded, continuous real-valued function on $mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of indicators of Borel sets
calculus real-analysis measure-theory reference-request borel-sets
edited Nov 26 at 23:57
T. Bongers
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
Where can I find a proof of the following fact?
Any bounded, continuous real-valued function on $mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of indicators of Borel sets
calculus real-analysis measure-theory reference-request borel-sets
calculus real-analysis measure-theory reference-request borel-sets
edited Nov 26 at 23:57
T. Bongers
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
edited Nov 26 at 23:57
T. Bongers
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
edited Nov 26 at 23:57
T. Bongers
22.8k54661
edited Nov 26 at 23:57
T. Bongers
22.8k54661
edited Nov 26 at 23:57
T. Bongers
22.8k54661
22.8k54661
asked Nov 26 at 23:37
user3503589
1,2011721
asked Nov 26 at 23:37
user3503589
1,2011721
asked Nov 26 at 23:37
user3503589
1,2011721
1,2011721
add a comment |
add a comment |
2 Answers
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A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=sum_j frac {j-1} n I_{f^{-1}(frac {j-1} n,frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| leq frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
add a comment |
I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.
I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate
$$f approx sum_{Q in mathcal{D}} chi_Q f_Q$$
where $mathcal{D}$ is a collection of disjoint sets $Q$ which cover $mathbb{R}^d$. (For example, $mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.
answered Nov 26 at 23:45
T. Bongers
22.8k54661
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=sum_j frac {j-1} n I_{f^{-1}(frac {j-1} n,frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| leq frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
add a comment |
A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=sum_j frac {j-1} n I_{f^{-1}(frac {j-1} n,frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| leq frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
add a comment |
A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=sum_j frac {j-1} n I_{f^{-1}(frac {j-1} n,frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| leq frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=sum_j frac {j-1} n I_{f^{-1}(frac {j-1} n,frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| leq frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
answered Nov 27 at 0:08
Kavi Rama Murthy
49.6k31854
49.6k31854
add a comment |
add a comment |
I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.
I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate
$$f approx sum_{Q in mathcal{D}} chi_Q f_Q$$
where $mathcal{D}$ is a collection of disjoint sets $Q$ which cover $mathbb{R}^d$. (For example, $mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.
answered Nov 26 at 23:45
T. Bongers
22.8k54661
add a comment |
I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.
I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate
$$f approx sum_{Q in mathcal{D}} chi_Q f_Q$$
where $mathcal{D}$ is a collection of disjoint sets $Q$ which cover $mathbb{R}^d$. (For example, $mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.
answered Nov 26 at 23:45
T. Bongers
22.8k54661
add a comment |
I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.
I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate
$$f approx sum_{Q in mathcal{D}} chi_Q f_Q$$
where $mathcal{D}$ is a collection of disjoint sets $Q$ which cover $mathbb{R}^d$. (For example, $mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.
answered Nov 26 at 23:45
T. Bongers
22.8k54661
I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.
I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate
$$f approx sum_{Q in mathcal{D}} chi_Q f_Q$$
where $mathcal{D}$ is a collection of disjoint sets $Q$ which cover $mathbb{R}^d$. (For example, $mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.
answered Nov 26 at 23:45
T. Bongers
22.8k54661
answered Nov 26 at 23:45
T. Bongers
22.8k54661
answered Nov 26 at 23:45
T. Bongers
22.8k54661
answered Nov 26 at 23:45
T. Bongers
22.8k54661
22.8k54661
add a comment |
add a comment |
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