I have got one question, because I see a big black hole in my knowledge about measure theory and Lebesgue's integrals:
$$lim \int_A \sqrt [n] {x_1x_2} dl_2(x_1 x_2), A = {x_1^2 + x_2^2 <1, 0 \le x_1 \le x_2}$$
and after that we found function g: $\sqrt [n] {x_1x_2} \le g(x_1,x_2) =1$ and next we are calculating integral g and it is sth like that: $\int _A 1 dl_2 = \frac{1}{8} \pi * 1^2$ it's like 1/8 of the field of disk. I have no idea why 1/8...?
Probably this is elementary question about Lebesgue's measure but I really can't discover how to calculate it.
Plot a graph of the conditions $x_1^2 + x_2^2 <1$ and $0 \le x_1 \le x_2$ in a plane whose coordinate axes are $x_1$ and $x_2$. The points that satisfy those two conditions simultaneously are just a segment of a disk, specifically, one eighth of a complete disk.