In my textbook, we have the following definition,
If $M$ is a smooth manifold of dimension $n$. We say that a set $A\subset M$ is measurable if, for any chart $U$, the intersection $A\cap U$ is a Lebesgue measurable set in $U$.
What really means this? A chart $U$ on $M$ is any couple $(U,\varphi)$ where $U$ is an open subset of $M$ and $\varphi$ is a homeomorphism of $U$ onto an open subset of $\mathbb{R}^{n}$. So, if for any chart $U$, the intersection $A\cap U\implies \varphi(A\cap U)\subset\mathbb{R}^{n}$ is Lesbesgue measurable set in $U$?
And, also says that
The family of all measurable sets in $M$ forms a $\sigma-$álgebra.
But I don't see how this form an $\sigma-$algebra, i.e., is the union of measurable set measurable?. Thanks!