Given x in $\displaystyle \mathbb{R^{N}}$ (real numbers of dimension N).

Write x = (x_{1},...,x_{N}) and x' = (x_{1},...,x_{N-1}), then we denote x = (x',x_{N}).

Let $\displaystyle \mathbb{R_{+}^{N}}$ = {x = (x', x_{N}): x_{N}> 0}

$\displaystyle \mathbb{R_{-}^{N}}$= {x = (x', x_{N}): x_{N}< 0}

Q = {x = (x',x_{N}): |x'| < 1 and |x_{N}| < 1}

Q_{+}= Q $\displaystyle \cap$ R^{N}_{+}

Q_{-}= Q $\displaystyle \cap$ R^{N}_{-}

Take u $\displaystyle \in$ L^{P}(Q_{+}), then we define a function u* to be an extension of u to all of Q by u*(x) = u(x',x_{N}) if x_{N}> 0 and

u*(x) = u(x',-x_{N}) if x_{N}< 0.

Then would it follow that if we use change of variable y = (x',-x_{N}) 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!