# Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure:

Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where all the functions $T_g$ are measurable.

A measure $\mu$ on $(X,\mathcal{B})$ is called conservative under $G$ if for every $A\in \mathcal{B}$ with $\mu(A) > 0$ exists $g \in G\setminus\{1\}$ with $$\mu(A\cap T_g^{-1}A) > 0$$ I was wondering if this definition is known in the literature or if it is specific to this paper?.

One can notice that if the action of $G$ is given by an invertible measurable function $T$ and the measure $\mu$ is ergodic, then by the von Neumann ergodic theorem (on $L^2(\mu)$) we have that for every $A \in \mathcal{B}$ with $\mu(A) > 0$ $$\frac{1}{n} \sum_{k=0}^{n-1} \mu(A \cap T^{-k}A) = \left\langle 1_A , \frac{1}{n} \sum_{k=0}^{n-1} 1_A \circ T^k \right\rangle_{L^2(\mu)} \stackrel{n \to \infty}{\longrightarrow} \mu(A)^2 > 0$$ So it must exists $k \in \mathbb{N}$ such that $$\mu(A \cap T^{-k}A) > 0$$ Hence $\mu$ is conservative by the group action $(T^n)_n$.

Edit: More generally if the measure is only invariant by the action $(T^n)_n$ we have the same result thanks to the Poincare Recurrence Theorem .

There is a more general connection between conservative measures and ergodic measures?

Any comment or reference will be appreciated.

• If $G \times X \to X, (g,x) \mapsto T_g(x)$ really is a group action, we could always take $g = 1_G$ to achieve $\mu(A \cap T_g^{-1}A) > 0$, couldn't we? – PhoemueX Oct 15 '14 at 21:31
• I forgot to put the restriction $g \in G \setminus \{1\}$. Thanks for your comment – user90803 Oct 15 '14 at 21:54

## 2 Answers

Conservative measures are a well known (old) concept coming from physics, where the system is conservative if it does not "loose" energy (e.g. frictionless). In this case a corresponding volume of the phase space is preserved and if one formalizes this notion we arrive at a conservative measure. Nevertheless this notion is a bit old-fashioned as people nowadays mostly talk about ergodic measures. Conservative measures have been kind of the predecessors of the notion of ergodic measures.

For a not that old reference (in the case of Z-actions), see the book Invitation to ergodic theory by Silva. Older books (from the 60s and 70s) on ergodic theory usually have this as well.

Conservativity of the action is not necessarily immediate for infinite measure spaces, and this is where this definition is useful (see Aaronson's book), for example $x\mapsto x+1$ is not conservative on $\mathbb{R}$ with the Haar measure (but certianly measure-preserving).

Maharam's theorem relates conservativity and recurrence in general.