Prove that fn is converge uniformly to 0
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In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?
measure-theory examples-counterexamples
asked Nov 26 at 16:47
Duaa Hamzeh
614
add a comment |
up vote
0
down vote
favorite
In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?
measure-theory examples-counterexamples
asked Nov 26 at 16:47
Duaa Hamzeh
614
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?
measure-theory examples-counterexamples
asked Nov 26 at 16:47
Duaa Hamzeh
614
In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?
measure-theory examples-counterexamples
measure-theory examples-counterexamples
asked Nov 26 at 16:47
Duaa Hamzeh
614
asked Nov 26 at 16:47
Duaa Hamzeh
614
asked Nov 26 at 16:47
Duaa Hamzeh
614
asked Nov 26 at 16:47
Duaa Hamzeh
614
asked Nov 26 at 16:47
Duaa Hamzeh
614
614
add a comment |
add a comment |
1 Answer
1
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oldest
votes
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1
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accepted
To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<epsilon$, which follows when $N=1/epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.
answered Nov 26 at 16:59
Richard Martin
1,6648
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<epsilon$, which follows when $N=1/epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.
answered Nov 26 at 16:59
Richard Martin
1,6648
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
add a comment |
up vote
1
down vote
accepted
To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<epsilon$, which follows when $N=1/epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.
answered Nov 26 at 16:59
Richard Martin
1,6648
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<epsilon$, which follows when $N=1/epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.
answered Nov 26 at 16:59
Richard Martin
1,6648
To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<epsilon$, which follows when $N=1/epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.
answered Nov 26 at 16:59
Richard Martin
1,6648
answered Nov 26 at 16:59
Richard Martin
1,6648
answered Nov 26 at 16:59
Richard Martin
1,6648
answered Nov 26 at 16:59
Richard Martin
1,6648
1,6648
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
add a comment |
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Thanks alot....
– Duaa Hamzeh
Nov 26 at 17:16
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
Does simple measurable function integrable every time?
– Duaa Hamzeh
Nov 26 at 17:18
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
OK so the first part of the qn says that if the range of integration is infinite then you can be in trouble.
– Richard Martin
Nov 26 at 17:19
add a comment |
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