Generated Finite Sub-Sigma-Algebra of Generated Finite Partition is Itself

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\BB\subseteq \Sigma$ be a finite sub-$\sigma$-algebra.

Then:

$\map \sigma {\map \xi \BB} = \BB$

where:

$\map \xi \cdot$ denotes the generated finite partition

$\map \sigma \cdot$ denotes the generated $\sigma$-algebra.

Proof

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Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras