# Existence of isometry-invariant measures

One of the wonderful things about the development of measure theory is that the most important theorems (Littlewood's principles, Dominated convergence, Fubini's theorem, etc.) can be proven in the context of a general measure space. The only extra properties we get out of Lebesgue measure on $\mathbb R^n$ is the fact that it is a Borel measure and its invariance under rigid motions (isometries). Inspired by this, I was wondering what conditions one would have to impose on a metric space in order to guarantee the existence of a Borel measure which is invariant under isometries. That is,

Desired Result: If (X,d) is a metric space satisfying condition $(P)$, then there exists a Borel measure $\mu$ on $X$ that is invariant under isometries, i.e. given an isometry $T:X\to X$, $$\mu(TB)=\mu(B)\textit{ for every Borel set }B\subset X$$

A neat example of a sufficient condition is given by the following theorem:

Theorem: Let $(X,d)$ be a locally compact metric topological group. If $d$ is left invariant, i.e. $$\forall x,y,z\in X:d(zx,zy)=d(x,y)$$ then a left-invariant Haar measure $\mu$ on $X$ is invariant under isometries.

For the proof of this theorem, see here: https://www.ams.org/journals/proc/1983-087-01/S0002-9939-1983-0677233-2/S0002-9939-1983-0677233-2.pdf (be warned: it's short, but involved!). Recall that this condition is sufficient, thanks to Haar's theorem:

Theorem: Let $G$ be a locally compact topological group. Then there exists a unique (modulo a multiplicative constant) left-invariant Haar measure $\mu$ on $G$. That is, $\mu$ has the property that $$\mu(gA)=\mu(A)\text{ for every Borel set }A\subset G$$

However, topological groups have a prior sort of invariance structure, namely, the action of group elements. The construction of Haar measure exploits this using the covering number $$(A:B)=\inf\{k:\exists x_1,\dots,x_k\in X,A\subset\cup_{i=1}^kx_iB\}$$ (for $A,B\subset X$) which yields a sufficient measure (pun intended) of relative size.

I have yet to find other references which prove the existence of such measures on a wider class of spaces. I suspect that the notion of 'relative size' given by group action might be totally integral to the construction of a measure, but I would love to be surprised!

Can anyone provide me a reference for other such existence proofs? Or better, does anyone know of a generalization of the above theorem?

I haven't looked at it in detail but the book by Fremlin has a section covering exactly what you want. You can find it on the author's website.

I'll just sketch some of the ideas here and refer to Fremlin's book for the details. In section 441.B the author discusses a theorem of Steinlage which is a generalization of the existence of Haar measures:

Theorem (Steinlage [1]): Let $X$ be a locally compact Hausdorff space and $G$ a group acting on $X$. Suppose that

1. the map $x \mapsto a \cdot x$ is continuous for every $a\in G$;
2. every orbit $\{ a\cdot x : a \in G \}$ is dense;
3. whenever $K$ and $L$ are disjoint compact subsets of $X$ there is a non-empty open subset $U \subset X$ such that, for every $a \in G$, at most one of $K$ or $L$ meets $a \cdot U$,

then there is a non-zero $G$-invariant Radon measure $\mu$ on $X$.

Taking the action of $G$ on itself by left/right translations, this theorem gives the existence of Haar measures. The idea of the proof is the same as the usual proof of the existence of Haar measures. First, for a compact set $K \subset X$ and an open set $U \subset X$ you define the ratio $[K:U]$ which is the minimal number of translates of $U$ needed to cover $K$. Then for a fixed reference compact set $K_0$ with non-empty interior you define the following function on compact subsets of $X$:

$$\lambda_U( K) = \frac{[K :U]}{[K_0 : U]}.$$ This is subadditive but not additive in general. To get an additive function you take some kind of limit over the open sets $U$ (here there are a lot of details to fill in of course!). Then you extend the function to an outer measure on $X$ by the usual sup/inf construction, and finally take the restriction of this outer measure to the Borel sigma algebra.

Returning to your question, in section 441.H Fremlin shows that the above theorem implies the following:

Theorem: Let $X$ be a locally compact metric space with isometry group $G$. Then there is a non-zero $G$-invariant Radon measure on $X$.

The only subtle thing to check here is condition 2 of Steinlage's theorem. There is no chance that it will hold in general for the full action of $G$ on $X$. So instead let's restrict to the closure of the orbit of a single point $x_0$, which we call $Z$, and look at the action of the isometry group $H$ of $Z$. For an arbitrary $z \in Z$, we have to prove that its orbit under $H$ is dense in $Z$. So let $y$ be an arbitrary point in $Z$. Then there is a sequence $(g_n)$ in $G$ such that $g_n x_0 \to y$. There is also a sequence $(h_n)$ in $H$ such that $h_n x_0 \to z$. Now since the metric is $G$-invariant, this implies that $g_n h_n^{-1} z \to y$. Indeed \begin{align} d(g_n h_n^{-1} z, y) &\leq d(g_n h_n^{-1} z, g_n x_0) + d(g_n x_0, y) \\ &= d(h_n^{-1} z, x_0) + d(g_n x_0, y) \\ &= d(z, h_n x_0) + d(g_n x_0, y), \end{align} which goes to $0$ as $n \to \infty$. Hence $y$ is in the closure of the orbit of $z$, and this shows that the $H$-orbit of any point is dense in $Z$.

Applying Steinlage's theorem we obtain an $H$-invariant measure $\nu$ on $Z$. To get a $G$-invariant measure on $X$ we define $\mu(E) = \nu(E \cap Z)$

[1] Steinlage, R. C. On Haar measure in locally compact T2 spaces. Amer. J. Math. 97 (1975), 291–307. Available here.