ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4437 on D2841: Stationary measurable set

Complement of stationary measurable set is stationary

Formulation 0

Let $M = (X, \mathcal{F}, \mu, T)$ be a D2827: Measure-preserving system such that

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} T^{-1}(E) = E \end{equation}

Then \begin{equation} T^{-1}(X \setminus E) = X \setminus E \end{equation}

Proofs

Proof 0

Let $M = (X, \mathcal{F}, \mu, T)$ be a D2827: Measure-preserving system such that

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} T^{-1}(E) = E \end{equation}

Using result R1530: Inverse image preserves set difference and the stationarity of $E$, we have \begin{equation} T^{-1}(X \setminus E) = T^{-1}(X) \setminus T^{-1}(E) = T^{-1}(X) \setminus E = X \setminus E \end{equation} $\square$