Baby Rudin section 10.28 on Affine Chains: if $\Gamma=\sigma_1+\sigma_2 = 0$ ,then why $\int_{\Gamma}\omega=0$?

Question

My question is this:

I don't understand if $\Gamma=\sigma_1+\sigma_2=0$ ,then why is it necessary for $\int_{\Gamma}$ $\omega$ to be equal to $0$?

Answer 1

I think you there are two ways to think about this. One is that integration is some kind of a summation, and empty sum is usually by definition 0. The other is if we denote by $C_n(E)$ the $n$-chains in $E$ which are just formal sum of maps from the standard $n$-simplex to $E$, and denote by $\Omega_n(E)$ the differential $n$-forms on $E$, then you should think about integration as a bilinear map $\int:C_n(E)\times \Omega_n(E)\rightarrow \mathbb{R}$. If you have $\Gamma=\sum_{i=1}^n a_i\sigma_i\in C_n(E)$ and $\omega\in \Omega_n(E)$, then $\int (\Gamma,\omega)=\sum_{i=1}^n a_i\int_{\sigma_i} \omega$. Now it is standard result in linear algebra that if $B$ is a bilinear form then $B(0,-)$ is the zero function.