Let
$$E := \{x \in \Bbb R^5: x_1, x_2 \in \Bbb R, x_3 = x_1 + x_2, x_4 = 1, x_5 = 2\}$$
a manifold in $\Bbb R^5$. Furthermore, let $\sigma$ be the surface-measure defined by the chart $\phi: \Bbb R^2 \rightarrow E$.
Show that
$$(E, B(E), \sigma)$$
is a measure space. At first, show that $(E, B(E))$ is a measurable space.
I know that there are various theorems for showing that a function is measurable, but I have never proven that something is a measurable space. I looked up the definition for it, and it seems like that I only have to show that $B(E)$ is a $\sigma$-algebra.
But how would I have to do that? I don't know anything about $B(E)$ in this case. The $B$ might imply that it has something to do with Borel-sets, but that's all about it.
Does anyone know how to start about it?