TITLE: how do we prove that a sum of two periods is still a period?
QUESTION [12 upvotes]: Kontsevich and Zagier define periods as the values of absolutely convergent integrals $\int_\sigma f$ where $f$ is a rational function with rational coefficients and $\sigma$ is a semi-algebraic subset of $\mathbb{R}^n$. How do we prove that the sum of two such numbers is still of this form? I've tried a few things but they don't seem to work...

REPLY [23 votes]: Let $\alpha$ and $\beta$ be two periods corresponding respectively to two absolutely convergent integrals $\int_\sigma f(x)dx$ and $\int_\tau g(y)dy$, where $f$ (resp. $g$) is a rational function on $\Bbb Q$ with $r$ (resp. $s$) variables and $\sigma$ (resp. $\tau$) is a semi-algebraic subset of $\Bbb R^r$ (resp. $\Bbb R^s$).
Setting $\omega:=\sigma\times(0,1)\times(0,1)^s\coprod(0,1)^r\times(1,2)\times\tau$, one immediately gets that $$\alpha+\beta=\int_\omega \left[f(x)+g(y)\right]dxdydt$$which is again an absolutely convergent integral, so that $\alpha+\beta$ is a period.