Jordan measurability of compact Half-space

Question

A half-space in $\mathbb{R}^d$ is of the form,$$\left\{ x\in \mathbb{R}^d:x\cdot v\leq c\text{ where, }v\in\mathbb{R}^d,c\in\mathbb{R}\right\}$$ how to prove a compact half space is Jordan measurable.It seems to me for $d=2$ case it is just a rotated rectangle{ a rectangle whose sides may not be parallel to coordinate axis} and hence Joran measurable, then what about the general case i.e $n>2$? it seems to me for the general case a compact half-space can be represented as the area under the graph of a continuous function with a compact domain, hence it is Jordan measurable. Am I right?