Norms, Metrics, Measures, and Volume on $\mathbb{R}^n$

Question

I want to understand better the interaction of norms (and metrics derived from them) and measure and volume in $\mathbb{R}^n$.

The volume or measure of a set on $\mathbb{R}^n$ is usually defined via the Lebesgue measure using the euclidean volume of cubes covering a set (the product of the length of the sides).

With this you end up with some nice properties like translation and rotation invariance of volume.

This is consistent with the Euclidean norm whose unit balls also share these properties.

However, with other norms on $\mathbb{R}^n$ this isn't the case. If we consider balls defined by the p-norms, $\left\lVert x\right\rVert_p$ with $p>2$, this is no longer the case.

If we take such a ball and then rotate it, it will have the same volume according to the Lebesgue measure but it no longer includes all points of unit distance from its center.

My Question

1. Why not define measures that are compatible with the norm (and the corresponding metric induced by the norm) over $\mathbb{R}^n$?

2. It seems that the Hausdorff Measure does this, i.e., assigning a fix volume to all unit balls based on the metric. The consequence of such a measure is that rotation invariance is no longer preserved.