Definition of integral on open set.

Question

I am learning about integration on manifolds. Suppose we have a $k$-form $\omega = a dx_1 \wedge \ldots \wedge dx_k$ where $a$ is some $0$-form on $U,$ an arbitrary real valued function. Then we define $\int_U \omega = \int_U a dx_1 \ldots dx_k.$ How is $\int_U a dx_1 \ldots dx_k$ defined?

Answer 1

It is defined as a regular $k-$variable function : $U$ is an open set in $\Bbb{R}^{k}$, and $a:\Bbb{R}^{k} \rightarrow \Bbb{R}$ so integration is well-defined here.