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This is a follow-up of two previous questions discussed:

Is every sigma-algebra generated by some random variable?

Can every filtration be written as $\mathcal F^X$ for some process $X$

Consider the standard Borel $\sigma$-algebra $\mathcal B(\mathbb R)$. Now, let's pick any non Borel measurable set $A$, and define the augmentation $$\mathcal F = \sigma (\mathcal B({\mathbb R}), A)$$

Question: Can such a $\sigma$-algebra be induced by a random variable? Or put in another way: Is it possible to find a map $f$ from $\mathbb R\to \mathbb R$, such that $$\mathcal F = \sigma (f^{-1}(S) | S\in \mathcal B({\mathbb R})\ )$$

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Jay.H
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1 Answers1

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How about $f(x):=1_A(x)+\big[{1\over 2}+{1\over\pi}\arctan(x)\big]$?

John Dawkins
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