equality between two sigma algebras

measure-theory

Solution 1:

For any subcollection $\mathcal{C}$ of $\wp\left(E_{1}\right)$ or $\wp\left(E_{2}\right)$ let $\sigma\left(\mathcal{C}\right)$ denote the $\sigma$-algebra generated by $\mathcal{C}$.

Instead of $\left(O_{i}\right)_{i\in I}$ I will write $\mathcal{V}$ so that $\sigma\left(\mathcal{V}\right)$ denotes the $\sigma$-algebra generated by $\mathcal{V}$.

Defining $f^{-1}\left(\mathcal{C}\right)=\left\{ f^{-1}\left(C\right)\mid C\in\mathcal{C}\right\} $ for any subcollection $\mathcal{C}$ of $\wp\left(E_{2}\right)$ you are actually asking whether the following statement is true: $$f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)=\sigma\left(f^{-1}\left(\mathcal{V}\right)\right)$$ The answer to that question is: "yes".

Start by proving the following statements:

  • $f^{-1}\left(\mathcal{A}\right)$ is a $\sigma$-algebra whenever $\mathcal{A}\subseteq\wp\left(E_{1}\right)$ is a $\sigma$-algebra.
  • $\left\{ A\in\wp\left(E_{2}\right)\mid f^{-1}\left(A\right)\in\mathcal{B}\right\} $ is a $\sigma$-algebra whenever $\mathcal{B}\subseteq\wp\left(E_{1}\right)$ is a $\sigma$-algebra.

They are not too difficult to prove, especially because preimages are quite nice to work with.

The first statement tells us immediately that $f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)$ is a $\sigma$-algebra, and this of course with $f^{-1}\left(\mathcal{V}\right)\subseteq f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)$.

This allows the conclusion that $\sigma\left(f^{-1}\left(\mathcal{V}\right)\right)\subseteq f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)$.

The second statement tells us that $\left\{ A\in\wp\left(E_{2}\right)\mid f^{-1}\left(A\right)\in\sigma\left(f^{-1}\left(\mathcal{V}\right)\right)\right\} $ is a $\sigma$-algebra and this with $\mathcal{V}\subseteq\left\{ A\in\wp\left(E_{2}\right)\mid f^{-1}\left(A\right)\in f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)\right\} $.

This allows the conclusion that $\sigma\left(\mathcal{V}\right)\subseteq\left\{ A\in\wp\left(E_{2}\right)\mid f^{-1}\left(A\right)\in\sigma\left(f^{-1}\left(\mathcal{V}\right)\right)\right\} $ or equivalently $f^{-1}\left(\sigma\left(\mathcal{V}\right)\right)\subseteq\sigma\left(f^{-1}\left(\mathcal{V}\right)\right)$.

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