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.
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.