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Why is the $delta$-approximative d-dimensional Hausdorff measure not $sigma$-additive?
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How can I show that the $delta$-approximative d-dimensional Hausdorff measure isn't a measure thus to say that it isn't $sigma$-additive?
hausdorff-measure
asked Jul 17 at 13:05
Rico1990
466
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up vote
-2
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How can I show that the $delta$-approximative d-dimensional Hausdorff measure isn't a measure thus to say that it isn't $sigma$-additive?
hausdorff-measure
asked Jul 17 at 13:05
Rico1990
466
1
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
2
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10
add a comment |Â
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
How can I show that the $delta$-approximative d-dimensional Hausdorff measure isn't a measure thus to say that it isn't $sigma$-additive?
hausdorff-measure
asked Jul 17 at 13:05
Rico1990
466
How can I show that the $delta$-approximative d-dimensional Hausdorff measure isn't a measure thus to say that it isn't $sigma$-additive?
hausdorff-measure
asked Jul 17 at 13:05
Rico1990
466
asked Jul 17 at 13:05
Rico1990
466
asked Jul 17 at 13:05
Rico1990
466
asked Jul 17 at 13:05
Rico1990
466
466
1
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
2
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10
add a comment |Â
1
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
2
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10
1
1
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
2
2
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10
add a comment |Â
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1
Please give more context: write the relevant definitions and your ideas about how to solve the problem.
– aduh
Jul 17 at 13:06
2
What is the $delta$-approximative $d$-dimensional Hausdorff measure of a singleton point? What about two distinct points that can be covered by a single $delta$-ball? What about two distinct points which cannot be covered by a single $delta$-ball?
– Xander Henderson
Jul 17 at 13:11
Thank you for your answer. For a singleton I suppose 0 since it covers itself. For a set of two singletons in the first case (let them be $x_1$ and $x_2$) $d(x_1,x_2)$ which is the metric derived form the euclidean norm. In the second case we can't find a cover so that we get $inf(emptyset)=infty$. All together gives us that we don't get equality for $H_delta^p(lbrace x_1,x_2 rbrace)$ and $H_delta^p(lbrace x_1 rbrace)+H_delta^p(lbrace x_2 rbrace)$.
– Rico1990
Jul 17 at 15:10