Absolutely continuous and differentiable almost everywhere
$begingroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
asked Sep 20 '18 at 18:21
user202542user202542
149311
$endgroup$
add a comment |
$begingroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
asked Sep 20 '18 at 18:21
user202542user202542
149311
$endgroup$
add a comment |
$begingroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
asked Sep 20 '18 at 18:21
user202542user202542
149311
$endgroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
integration derivatives bochner-spaces absolute-continuity
asked Sep 20 '18 at 18:21
user202542user202542
149311
asked Sep 20 '18 at 18:21
user202542user202542
149311
asked Sep 20 '18 at 18:21
user202542user202542
149311
asked Sep 20 '18 at 18:21
user202542user202542
149311
asked Sep 20 '18 at 18:21
user202542user202542
149311
149311
add a comment |
add a comment |
1 Answer
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$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
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1 Answer
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1 Answer
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$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
$endgroup$
add a comment |
$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
$endgroup$
add a comment |
$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
$endgroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
2,5363920
add a comment |
add a comment |
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