**The general question**

It is easy to find on the Wikipedia page for Lebesgue measure that Haar measure is a common generalization that preserves the idea of "invariance under some group action". While wondering about the "most natural" way of defining a measure on lines of $\mathbb{R}^2$ (see below for more information), it struck me that it's not always obvious what the "most natural" measure is for certain spaces is like in $\mathbb{R}^n$. I was wondering how this problem is usually tackled, if it even comes up.

Is finding an "appropriate" Haar measure usually easy to find? Is it unique in some sense or usually dependent on the use case? What can we do in the case of the space of lines in $\mathbb{R}^2$ (or general hyperplanes in $\mathbb{R}^n$)?

**The specific case of lines in $\mathbb{R}^2$**

To give an example, and some motivation, consider lines in $\mathbb{R}^2$. Is there a "unique", "most natural" measure on this space? By this, I mean some invariance under isometries, plus some extra properties that satisfy my intuition of what it should look like. My primary motivation is whether there is actually a "most natural" way to solve Bertrand's paradox.

Specifically something satisfying the properties below would be nice. In what follows, $S$ is a measurable set of lines, $m(S)$ is the measure of $S$, and if we have an operation $f:\mathbb{R}^2\to \mathbb{R}^2$ we can consider $f(S)$ to be the set of $f(L)$ for each $L$ in $S$ (where $f(L)$ for some line $L$ is just the set of $f(p)$ for $p$ in $L$).

- If $f$ is a rotation, $m(S)=m(f(S))$
- If $f$ is a reflection, $m(S)=m(f(S))$
- If $f$ is a translation, $m(S)=m(f(S))$
- If $f$ is a dilation with factor $C$ (i.e.- two points $d$ apart get dilated to $C\cdot d$ apart), then $C^2\cdot m(S)=m(f(S))$
- If $M$ is a compact subset of $\mathbb{R}^2$, then the set of lines $S$ that intersect $M$ is measurable and $0<m(S)<∞$

I see three cases: Either there are no measures satisfying the above, in which case perhaps there is a weakening of the conditions to give some meaningful content (we might be able to at least satisfy a chosen subset of the above); there is exactly one measure satisfying the above (up to perhaps some relatively trivial modifications) and I am happy; or there are many and perhaps we need some more conditions specifying what a "natural" measure should be.

Of course, this is a relatively open-ended question, so I am satisfied with any related comments. I simply don't want to wade through a couple courses in measure theory just for a chance of learning the answer. :)

not requiredto pass through the origin, which is not the Grassmannian (and noncompact). After all, the Grassmannian of lines through the origin for $\mathbb{R}^2$ is just the circle (with its usual rotationally invariant measure). $\endgroup$ – Robert Furber May 17 at 15:464more comments