If $f(\mathbf{x}): R^p \rightarrow R $ is a multivariate odd-symmetric function in the sense that $f(\mathbf{x}) = -f(-\mathbf{x})$ for any $\mathbf{x}$ and it is absolutely integrable, does it integrate to 0 on $R^p$?
Any references will be appreciated. Thank you!
The multivariate odd symmetric definition was found on https://en.wikipedia.org/wiki/Even_and_odd_functions
How about doing the substiution $y=-x$,
$$\int_{\mathbb R^p} f(x) \mathrm dx= \int_{\mathbb R^p} f(-y) \mathrm dy = -\int_{\mathbb R^p} f(y)\mathrm dy$$
Can you finish the proof?