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

Result R4469 on D2841: Stationary measurable set

Countable union of stationary measurable sets is stationary

Formulation 0

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

(i)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each a D2841: Stationary measurable set in $M$

Then \begin{equation} T^{-1} \left( \bigcup_{n \in \mathbb{N}} E_n \right) = \bigcup_{n \in \mathbb{N}} E_n \end{equation}

Proofs

Proof 0

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

(i)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each a D2841: Stationary measurable set in $M$

Since $E_0, E_1, E_2, \ldots$ are all stationary in $M$, applying R4470: Inverse image of countable union is union of inverse images, we have \begin{equation} T^{-1} \left( \bigcup_{n \in \mathbb{N}} E_n \right) = \bigcup_{n \in \mathbb{N}} T^{-1} E_n = \bigcup_{n \in \mathbb{N}} E_n \end{equation} $\square$