Let $f:(X,\mathbb{A})\rightarrow\mathbb{R}$ a defined function in a measurable space... Prove...

Question

Let $C$ a class such that $C$ generate $B(\mathbb{R})$ (Borel $\sigma$-Algebra).
Let $f:(X,\mathbb{A})\rightarrow\mathbb{R}$ a defined function in a measurable space a real values, then

$$f^{-1}(B(\mathbb{R}))\subset\mathbb{A}\iff f^{-1}(C)\subset\mathbb{A}$$

Note we have:

$$\sigma(f^{-1}(C))=f^{-1}(\sigma(C))$$

As $C$ generate $B(\mathbb{R})$ then

$$\sigma(C)=B(\mathbb{R})$$

then

$$\sigma(f^{-1}(C))=f^{-1}(B(\mathbb{R}))\subset\mathbb{A}$$

this because $f^{-1}(C)\subset \sigma(f^{-1}(C)).$

Is correct this?

For the other implications i don't have a clear idea

What have you tried? Its an iff statement so there are two statements to prove. Have you tried either of them? Have you expanded all the terms out to the base definitions? — Calvin Khor, Jun 24 '19 at 16:30
Do you mean Borel $\sigma$-algebra? I am not familiar with "Borel Algebra". — drhab, Jun 24 '19 at 16:39

score 1 · Accepted Answer · answered Jun 24 '19 at 16:48

Your idea is okay (for a proof of it see here) and can be applied to prove that: $$f^{-1}(B(\mathbb R))=f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$

So the statement can be rewritten as: $$\sigma(f^{-1}(C))\subseteq\mathbb A\iff f^{-1}(C)\subseteq\mathbb A$$

Here ($\implies$) follows from $f^{-1}(C)\subseteq\sigma(f^{-1}(C))$ and ($\impliedby$) can be concluded from the fact that $\mathbb A$ is a $\sigma$-algebra.

Let $f:(X,\mathbb{A})\rightarrow\mathbb{R}$ a defined function in a measurable space... Prove...

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