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As the title says, I wonder if it is possible to calculate a Lebesgue integral in MathematicaMathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of MathematicaMathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.

Note: Why am I asking such a question here?

(i) I checked google about it and found no answer

(ii) I checked the Mathematica help and found no result

(iii) I searched the keyword "Lebesgue" among thequestions in mathematica.stackexchange.com but couldnt see any related answer

(iiii) I have no idea how I can do it in Mathematica

  • I checked google about it and found no answer
  • I checked the Mathematica help and found no result
  • I searched the keyword "Lebesgue" among the questions in mathematica.stackexchange.com but couldnt see any related answer
  • I have no idea how I can do it in Mathematica

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.

Note: Why am I asking such a question here?

(i) I checked google about it and found no answer

(ii) I checked the Mathematica help and found no result

(iii) I searched the keyword "Lebesgue" among thequestions in mathematica.stackexchange.com but couldnt see any related answer

(iiii) I have no idea how I can do it in Mathematica

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Note: Why am I asking such a question here?

  • I checked google about it and found no answer
  • I checked the Mathematica help and found no result
  • I searched the keyword "Lebesgue" among the questions in mathematica.stackexchange.com but couldnt see any related answer
  • I have no idea how I can do it in Mathematica
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As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.

Note: Why am I asking such a question here?

(i) I checked google about it and found no answer

(ii) I checked the Mathematica help and found no result

(iii) I searched the keyword "Lebesgue" among thequestions in mathematica.stackexchange.com but couldnt see any related answer

(iiii) I have no idea how I can do it in Mathematica

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.

Note: Why am I asking such a question here?

(i) I checked google about it and found no answer

(ii) I checked the Mathematica help and found no result

(iii) I searched the keyword "Lebesgue" among thequestions in mathematica.stackexchange.com but couldnt see any related answer

(iiii) I have no idea how I can do it in Mathematica

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Is it possible to calculate a Lebesgue integral in Mathematica?

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue integration is of interest. As an example, one could take $f_0$ and $f_1$ as two different (possibly correlated) bi-variate Gaussian density functions and consider

$$\int_{\frac{f_1}{f_0}>\tau}f_0\mbox{d}\mu=\int_{\Large\{x,y:\frac{f_1(x,y)}{f_0(x,y)}>\tau\Large\}}f_0(x,y)\mbox{d}x\mbox{d}y$$ for some known number $\tau$.

If it is not possible can one manipulate the existent functions of Mathematica to get a method which can calcuate the Lebesgue integral. I am only interested in very fast numerical methods, no analytical results are needed.

Thank you very much.