Given x in (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 = {x = (x', x_{N}): x_{N} > 0}
= {x = (x', x_{N}): x_{N} < 0}
Q = {x = (x',x_{N}): |x'| < 1 and |x_{N}| < 1}
Q_{+} = Q R^{N}_{+}
Q_{-} = Q R^{N}_{-}
Take u 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:
?
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!