L1 as Probability space
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In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $lbrack 0,1 rbrack^N $ that admit a probability density. I understand that I exclude in the process distributions such as the Dirac.
Can I use a subset of the Lebesgue $L_1(lbrack 0,1 rbrack^N) $ space for that matter?
To be specific, can I use :
$$ { p in L_1 | int p = 1, pgeq0 } $$
probability probability-theory probability-distributions
asked Nov 20 at 8:51
Mathieu
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0
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In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $lbrack 0,1 rbrack^N $ that admit a probability density. I understand that I exclude in the process distributions such as the Dirac.
Can I use a subset of the Lebesgue $L_1(lbrack 0,1 rbrack^N) $ space for that matter?
To be specific, can I use :
$$ { p in L_1 | int p = 1, pgeq0 } $$
probability probability-theory probability-distributions
asked Nov 20 at 8:51
Mathieu
12
Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $lbrack 0,1 rbrack^N $ that admit a probability density. I understand that I exclude in the process distributions such as the Dirac.
Can I use a subset of the Lebesgue $L_1(lbrack 0,1 rbrack^N) $ space for that matter?
To be specific, can I use :
$$ { p in L_1 | int p = 1, pgeq0 } $$
probability probability-theory probability-distributions
asked Nov 20 at 8:51
Mathieu
12
In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $lbrack 0,1 rbrack^N $ that admit a probability density. I understand that I exclude in the process distributions such as the Dirac.
Can I use a subset of the Lebesgue $L_1(lbrack 0,1 rbrack^N) $ space for that matter?
To be specific, can I use :
$$ { p in L_1 | int p = 1, pgeq0 } $$
probability probability-theory probability-distributions
probability probability-theory probability-distributions
asked Nov 20 at 8:51
Mathieu
12
asked Nov 20 at 8:51
Mathieu
12
edited Nov 20 at 9:53
asked Nov 20 at 8:51
Mathieu
12
asked Nov 20 at 8:51
Mathieu
12
asked Nov 20 at 8:51
Mathieu
12
12
Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45
add a comment |
Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45
Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45
add a comment |
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Yes,you can. This identification is useful in many ways.
– Kavi Rama Murthy
Nov 20 at 8:53
This is not fully accurate. What is the underlying space of $L_1$? What measure are integrating the function $p$ against?
– uniquesolution
Nov 20 at 9:04
Thanks both of you for your answers. $L_1$ is defined on $lbrack 0, 1 rbrack^N$ where N is an integer, and the measure is the Lebesgue measure. I will edit the question.
– Mathieu
Nov 20 at 9:45