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.