Given x in $\displaystyle \mathbb{R^{N}}$ (real numbers of dimension N).
Write x = (x1,...,xN) and x' = (x1,...,xN-1), then we denote x = (x',xN).

Let $\displaystyle \mathbb{R_{+}^{N}}$ = {x = (x', xN): xN > 0}
$\displaystyle \mathbb{R_{-}^{N}}$= {x = (x', xN): xN < 0}
Q = {x = (x',xN): |x'| < 1 and |xN| < 1}
Q+ = Q $\displaystyle \cap$ RN+
Q- = Q $\displaystyle \cap$ RN-

Take u $\displaystyle \in$ 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:
$\displaystyle \int_{Q_{-}} u(x',-x_{N})dx = \int_{Q_{+}}u(y',y_{N})dy$ ?

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.

Thanks!