Take u LP(Q+), then we define a function u* to be an extension of u to all of Q by u*(x) = u(x',xN) if xN > 0 and
u*(x) = u(x',-xN) if xN < 0.
Then would it follow that if we use change of variable y = (x',-xN) then: ?
If so why?
Does it follow from the fact that for Lebesgue integration the fact that the measurable
sets in Q+ and Q- have the same measures is all that is important and the orientation
has no consequence, or does it follow from a specific formula used to motivate why we can change our integration
from being over Q- to Q+?
Let me know if anything is unclear.