Mixing
Converging almost everywhere in finite space but not in p-th mean
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Give example of a finite measure space and a sequence of functions which converges $mu$-a.e but not in p-mean, for any $pgeqslant 1$.
I was thinking of the following spaces $([0,1],mathscr{B_{[0,1]}},lambda)$ and $(mathbb{R},mathscr{B_{mathbb{R}}},lambda)$. I need to find a sequence of function that as $nto 0$ the function goes to infinity.
I was tinking of $f(x)=x^n$ for $0<x<1$. But I am not sure.
Question:
Which function would you suggest?
Thanks in advance!
functional-analysis measure-theory examples-counterexamples
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
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add a comment |
$begingroup$
Give example of a finite measure space and a sequence of functions which converges $mu$-a.e but not in p-mean, for any $pgeqslant 1$.
I was thinking of the following spaces $([0,1],mathscr{B_{[0,1]}},lambda)$ and $(mathbb{R},mathscr{B_{mathbb{R}}},lambda)$. I need to find a sequence of function that as $nto 0$ the function goes to infinity.
I was tinking of $f(x)=x^n$ for $0<x<1$. But I am not sure.
Question:
Which function would you suggest?
Thanks in advance!
functional-analysis measure-theory examples-counterexamples
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
$endgroup$
add a comment |
$begingroup$
Give example of a finite measure space and a sequence of functions which converges $mu$-a.e but not in p-mean, for any $pgeqslant 1$.
I was thinking of the following spaces $([0,1],mathscr{B_{[0,1]}},lambda)$ and $(mathbb{R},mathscr{B_{mathbb{R}}},lambda)$. I need to find a sequence of function that as $nto 0$ the function goes to infinity.
I was tinking of $f(x)=x^n$ for $0<x<1$. But I am not sure.
Question:
Which function would you suggest?
Thanks in advance!
functional-analysis measure-theory examples-counterexamples
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
$endgroup$
Give example of a finite measure space and a sequence of functions which converges $mu$-a.e but not in p-mean, for any $pgeqslant 1$.
I was thinking of the following spaces $([0,1],mathscr{B_{[0,1]}},lambda)$ and $(mathbb{R},mathscr{B_{mathbb{R}}},lambda)$. I need to find a sequence of function that as $nto 0$ the function goes to infinity.
I was tinking of $f(x)=x^n$ for $0<x<1$. But I am not sure.
Question:
Which function would you suggest?
Thanks in advance!
functional-analysis measure-theory examples-counterexamples
functional-analysis measure-theory examples-counterexamples
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
asked Jan 30 at 12:02
Pedro GomesPedro Gomes
1,9872721
1,9872721
add a comment |
add a comment |
1 Answer
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On the space you are considering take $f_n(x)=n$ of $0<x<frac 1 n$ and $0$ otherwise.
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
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$begingroup$
On the space you are considering take $f_n(x)=n$ of $0<x<frac 1 n$ and $0$ otherwise.
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
$endgroup$
add a comment |
$begingroup$
On the space you are considering take $f_n(x)=n$ of $0<x<frac 1 n$ and $0$ otherwise.
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
$endgroup$
add a comment |
$begingroup$
On the space you are considering take $f_n(x)=n$ of $0<x<frac 1 n$ and $0$ otherwise.
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
$endgroup$
On the space you are considering take $f_n(x)=n$ of $0<x<frac 1 n$ and $0$ otherwise.
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
answered Jan 30 at 12:04
Kavi Rama MurthyKavi Rama Murthy
72.1k53170
72.1k53170
add a comment |
add a comment |
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