When do Inverse Images of Measurable Sets Generate the $\sigma$-algebra?

Question

Just out of curiosity!

Let $(X,\mathcal{M})$ and $(Y,\mathcal{N})$ be measurable spaces and let $f: X\to Y$ be a measurable function. If $\mathcal{N}$ is generated by some collection of sets $\mathcal{E} \subset \mathcal{P}(Y),$ when can we say that $\mathcal{M}$ is generated by $f^{-1}(\mathcal{E}),$ where $f^{-1}(\mathcal{E}) = \left\{f^{-1} (E) \mid E \in \mathcal{E}\right\}?$

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

2

According to this thread, we have $$\sigma(f^{-1}(\mathcal E))=f^{-1}(\sigma(\mathcal E))=f^{-1}(\mathcal N)\subset \mathcal M,$$ but the inclusion may be strict, for example when $f$ is constant.

edited Apr 13 '17 at 12:20

Community

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answered Sep 15 '13 at 20:13

Davide Giraudo

172,925

When do Inverse Images of Measurable Sets Generate the $\sigma$-algebra?

1 Answers1