M cannot be shown to be a monotone class. Because it is not easy to prove that M is closed under countable decreasing intersections because the Lebesgue measure is not finite and hence we can't say that it is continuous from above. To prove the required argument i.e. Lebesgue measure is translation invariant one can use Caratheodory's extension theorem. BTW how can you say that Lebesgue measure of E_1 is finite while proving continuity of the Lebesgue measure from above??
is it true that our usual double integral over region of R^2 (which is in some text given as "definition") is just lebesgue integral with respect to product measure on R^2?
M cannot be shown to be a monotone class. Because it is not easy to prove that M is closed under countable decreasing intersections because the Lebesgue measure is not finite and hence we can't say that it is continuous from above. To prove the required argument i.e. Lebesgue measure is translation invariant one can use Caratheodory's extension theorem.
BTW how can you say that Lebesgue measure of E_1 is finite while proving continuity of the Lebesgue measure from above??
is it true that our usual double integral over region of R^2 (which is in some text given as "definition") is just lebesgue integral with respect to product measure on R^2?