Lebesgue integral over $\{(x,y,z)\in \mathbb R^3:(x-\chi_\mathbb Q(z))^2+(y-e^z)^2\leq 3\sin(\pi z), z\in[0,1]\}$

Question

How can I calculate $\int_E 1d\lambda$
for $E:=\{(x,y,z)\in \mathbb R^3:(x-\chi_\mathbb Q(z))^2+(y-e^z)^2\leq 3\sin(\pi z), z\in[0,1]\}$

score 0 · Answer 1 · answered May 31 '22 at 11:05

0

You can split up the set, into $z \in [0, 1]\setminus \mathbb{Q}$ and $z \in [ 0, 1]\cap \mathbb{Q}$. With this you get rid of the charcteristic function. (Unfortunately I can't comment yet)

answered May 31 '22 at 11:05

Richard

575

Lebesgue integral over $\{(x,y,z)\in \mathbb R^3:(x-\chi_\mathbb Q(z))^2+(y-e^z)^2\leq 3\sin(\pi z), z\in[0,1]\}$

1 Answers1