Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable homeomorphism. For each $g\in G$, we use $T_g$ to denote the mapping $x\mapsto g\cdot x$. For each $g\in G$, we further define $g\cdot\mu$ by, for each $B\in\Omega$, $(g\cdot\mu)(B) = \mu(T_g^{-1}(B))$. Now assume each $g\cdot\mu$ is equivalent to $\mu$ (or the group action is non-singular in some textbooks). My questions are:
Is there a necessary condition for the following to be true: for each non-null $A\in\Omega$, the following set: $$ \Big\{ a\in A\,\vert\,\frac{d\,(g\cdot\mu)}{d\,\mu}(a) \in \Big( \frac{(g\cdot\mu)(A)}{\mu(A)}-\epsilon, \frac{(g\cdot\mu)(A)}{\mu(A)}+\epsilon \Big) \Big\}\tag{$\ast$}$$ is also non-null for each $\epsilon>0$ ? If it is non-null, could its measure be arbitrarily close to $\mu(A)$ or $0$?
This question is inspired by the first one, and related to a dynamical system I am working on. Now let's turn to a one-sided Markov shift $(X, \mathcal{A}, \mu, T)$ where $X = \{1, 2, \cdots, n\}^{\mathbb{N}}$, $T$ is the shift function and $\mathcal{A}$ is the $\sigma$ generated by finite cylinder sets. $\mu$ is defined by a stochastic $n\times n$ matrix $M = [m_{i, j}]$ and an initial distribution given by a row vector $\nu = [\nu_1, \cdots, \nu_n]$ such that, for instance, for the finite cylinder set $[x_1 = n_1, \cdots, x_k = n_k]$ (where $\{n_1, \cdots, n_k\}\subseteq X$), we have: $$ \mu \Big( [x_1=n_1, \cdots, x_k=n_k] \Big) = \nu_1\prod_{1\leq i < k} m_{n_i, n_{i+1}}$$ and we further assume $\nu M$ is equivalent to $\nu$. With these conditions, we then have, for each $A\in\Omega, \mu(A)=0$ iff $\mu[T^{-1}(A)]=0$ and, for each $n\in\mathbb{N}$, the measure $T^n\cdot\mu$ defined by $T^n\cdot\mu(A) = \mu[(T^n)^{-1}(A)]$ is equivalent to $\mu$. Now my question is, is the set in $(\ast)$ non-null in $\mathcal{A}$ for each $n\in\mathbb{N}$ and for all $\epsilon>0$, with $g\cdot\mu$ replaced by $T^n\cdot\mu$, and with $A$ from $\mathcal{A}$?
For the second question, the set $\Big\{ (T^n\cdot\mu)(A)/\mu(A)\,\vert\,n\in\mathbb{N}, A\text{ is a finite cylinder set} \Big\}$ is the multiplicative group generated by non-zero entries of $M$. If $\{A_i\}$ is a sequence of finite cylinder sets such that $\mu(A_i\Delta A)\rightarrow 0$, I am not sure, for a fixed $n\in\mathbb{N}$, if $\{(T^n\cdot\mu)(A_i)/\mu(A_i)\}$ will also approach to a number or diverge. For the first question, it seems like even the set in $(\ast)$ is null, still $\int_A \frac{d\,(g\cdot\mu)}{d\,\mu}\,d\mu$ can be equal to $(g\cdot\mu)(A)$. Any hints to either question will be appreciated.
