
Lebesgue Integration
Given x in $\displaystyle \mathbb{R^{N}}$ (real numbers of dimension N).
Write x = (x_{1},...,x_{N}) and x' = (x_{1},...,x_{N1}), 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!