Estimating Lebesgue measure via discrete points
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I am faced with the following problem, I need to estimate the Lebesgue measure of the level set ${f>alpha}$ within a program. The function $f$ is approximated by a vector of arguments $(t_i)_{i=1}^n$ and a vector of corresponding values $(y_i)_{i=1}^n$. The only solution that comes to mind is to approximate the measure by $hat{lambda}=frac{1}{n}|{t_i:y_i>alpha}|$. This raised an interesting question for me. Given a measurable set $Asubset [0,1]$ and an equidistant partition $t_i=frac{i}{n}$ with $i=0,ldots,n$, I define the estimator $hatlambda_n(A):=frac{1}{n}|{t_i:t_iin A}|$. What conditions must I impose on $A$ such that
$$
lim_{ntoinfty}hatlambda_n(A)=lambda(A),, ?
$$
Also, if anyone has a suggestion on how to better approximate the measure of a level set given a discrete approximation, I would be very glad to hear from you. Thank you in advance
measure-theory lebesgue-measure programming
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
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up vote
3
down vote
favorite
I am faced with the following problem, I need to estimate the Lebesgue measure of the level set ${f>alpha}$ within a program. The function $f$ is approximated by a vector of arguments $(t_i)_{i=1}^n$ and a vector of corresponding values $(y_i)_{i=1}^n$. The only solution that comes to mind is to approximate the measure by $hat{lambda}=frac{1}{n}|{t_i:y_i>alpha}|$. This raised an interesting question for me. Given a measurable set $Asubset [0,1]$ and an equidistant partition $t_i=frac{i}{n}$ with $i=0,ldots,n$, I define the estimator $hatlambda_n(A):=frac{1}{n}|{t_i:t_iin A}|$. What conditions must I impose on $A$ such that
$$
lim_{ntoinfty}hatlambda_n(A)=lambda(A),, ?
$$
Also, if anyone has a suggestion on how to better approximate the measure of a level set given a discrete approximation, I would be very glad to hear from you. Thank you in advance
measure-theory lebesgue-measure programming
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I am faced with the following problem, I need to estimate the Lebesgue measure of the level set ${f>alpha}$ within a program. The function $f$ is approximated by a vector of arguments $(t_i)_{i=1}^n$ and a vector of corresponding values $(y_i)_{i=1}^n$. The only solution that comes to mind is to approximate the measure by $hat{lambda}=frac{1}{n}|{t_i:y_i>alpha}|$. This raised an interesting question for me. Given a measurable set $Asubset [0,1]$ and an equidistant partition $t_i=frac{i}{n}$ with $i=0,ldots,n$, I define the estimator $hatlambda_n(A):=frac{1}{n}|{t_i:t_iin A}|$. What conditions must I impose on $A$ such that
$$
lim_{ntoinfty}hatlambda_n(A)=lambda(A),, ?
$$
Also, if anyone has a suggestion on how to better approximate the measure of a level set given a discrete approximation, I would be very glad to hear from you. Thank you in advance
measure-theory lebesgue-measure programming
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
I am faced with the following problem, I need to estimate the Lebesgue measure of the level set ${f>alpha}$ within a program. The function $f$ is approximated by a vector of arguments $(t_i)_{i=1}^n$ and a vector of corresponding values $(y_i)_{i=1}^n$. The only solution that comes to mind is to approximate the measure by $hat{lambda}=frac{1}{n}|{t_i:y_i>alpha}|$. This raised an interesting question for me. Given a measurable set $Asubset [0,1]$ and an equidistant partition $t_i=frac{i}{n}$ with $i=0,ldots,n$, I define the estimator $hatlambda_n(A):=frac{1}{n}|{t_i:t_iin A}|$. What conditions must I impose on $A$ such that
$$
lim_{ntoinfty}hatlambda_n(A)=lambda(A),, ?
$$
Also, if anyone has a suggestion on how to better approximate the measure of a level set given a discrete approximation, I would be very glad to hear from you. Thank you in advance
measure-theory lebesgue-measure programming
measure-theory lebesgue-measure programming
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
asked Nov 8 at 12:07
Nicolas Bourbaki
7511612
7511612
I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41
add a comment |
I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41
I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41
add a comment |
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I don't think you can do anything for arbitrary sets. You need to consider level sets of functions with some regularity. Otherwise, you will always run into sets such as $mathbb Qcap [0, 1]$, which has measure $0$, but is detected as a set of measure $1$ by your discrete estimator.
– Giuseppe Negro
Nov 15 at 15:34
Would a condition be that $A$ is a union of open intervals?
– Elsa
Nov 15 at 16:31
It would suffice to assume that the topological boundary of $A$ has measure $0$. As to the approximation of the measure of the level set, you may do some interpolation first and then compute or estimate the measure of the level set of the result. How to interpolate depends on what kind of function you expect $f$ to be.
– fedja
Nov 19 at 16:41