Convergence of Bounded Variation on Open Subsets
$begingroup$
I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $mathbb{R}^n$
Let $U_k := {x in U,|, text{dist}(x, partial Omega) > frac{1}{k} cap B_k(0)}$ where $B_k(0)$ is the ball of radius $k$ centered at $0$. Then $$
||Df||(U) := sup left{int_{U} f mathrm{ div } varphi; |; varphi in C_c^{infty}(U), |varphi| leq 1right };text{ implies };||Df||(U-U_k) rightarrow 0.
$$
It is easy to verify this for $f in W^{1,1}(U)$ since we have
$$
||Df||(U) = int_{U} |f'| dx.
$$
But for the general case, I'm having problems verifying this lemma. Any help would be appreciated.
real-analysis measure-theory bounded-variation
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
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add a comment |
$begingroup$
I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $mathbb{R}^n$
Let $U_k := {x in U,|, text{dist}(x, partial Omega) > frac{1}{k} cap B_k(0)}$ where $B_k(0)$ is the ball of radius $k$ centered at $0$. Then $$
||Df||(U) := sup left{int_{U} f mathrm{ div } varphi; |; varphi in C_c^{infty}(U), |varphi| leq 1right };text{ implies };||Df||(U-U_k) rightarrow 0.
$$
It is easy to verify this for $f in W^{1,1}(U)$ since we have
$$
||Df||(U) = int_{U} |f'| dx.
$$
But for the general case, I'm having problems verifying this lemma. Any help would be appreciated.
real-analysis measure-theory bounded-variation
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
$endgroup$
add a comment |
$begingroup$
I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $mathbb{R}^n$
Let $U_k := {x in U,|, text{dist}(x, partial Omega) > frac{1}{k} cap B_k(0)}$ where $B_k(0)$ is the ball of radius $k$ centered at $0$. Then $$
||Df||(U) := sup left{int_{U} f mathrm{ div } varphi; |; varphi in C_c^{infty}(U), |varphi| leq 1right };text{ implies };||Df||(U-U_k) rightarrow 0.
$$
It is easy to verify this for $f in W^{1,1}(U)$ since we have
$$
||Df||(U) = int_{U} |f'| dx.
$$
But for the general case, I'm having problems verifying this lemma. Any help would be appreciated.
real-analysis measure-theory bounded-variation
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
$endgroup$
I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $mathbb{R}^n$
Let $U_k := {x in U,|, text{dist}(x, partial Omega) > frac{1}{k} cap B_k(0)}$ where $B_k(0)$ is the ball of radius $k$ centered at $0$. Then $$
||Df||(U) := sup left{int_{U} f mathrm{ div } varphi; |; varphi in C_c^{infty}(U), |varphi| leq 1right };text{ implies };||Df||(U-U_k) rightarrow 0.
$$
It is easy to verify this for $f in W^{1,1}(U)$ since we have
$$
||Df||(U) = int_{U} |f'| dx.
$$
But for the general case, I'm having problems verifying this lemma. Any help would be appreciated.
real-analysis measure-theory bounded-variation
real-analysis measure-theory bounded-variation
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
edited Jan 26 at 7:35
Daniele Tampieri
2,3272922
2,3272922
asked Jan 23 at 11:19
User12335425User12335425
211
asked Jan 23 at 11:19
User12335425User12335425
211
asked Jan 23 at 11:19
User12335425User12335425
211
211
add a comment |
add a comment |
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