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Let $X=\mathbb{R}$ and $S=\{A\subset\mathbb{R}:A\space \text{or}\space A^{c}\space \text{is finite or countable}\}.$ Describe functions $f:X\rightarrow\overline{\mathbb{R}}$ what are $S-$mesurables.

I'm stuck on this; if $f$ is $S-$measurable then any Borelian set is inverse image of a finite or countable set of $S,$ but I don't know how to characterize them.

Any kind of help is thanked in advanced.

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    $\begingroup$ No you got the definition of measurable functions a little wrong, you should try to prove that the inverse image of any borel set is an element of $S$. $\endgroup$ – Abishanka Saha Dec 30 '17 at 2:33
  • $\begingroup$ See here. $\endgroup$ – Michael Greinecker Dec 30 '17 at 10:09
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Hint: If you cover $\mathbb R$ by (countably many) intervals of finite length, then you can find one with uncountable (-ly infinite) preimage under $f$ (an $S$-measurable function). Next you can find a smaller interval with uncountable preimage...

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  • $\begingroup$ Thanks for answer @Ben. So, as the hint, any interval $I\subset\mathbb{R}$ has the same characterization but, what about, for example, single point $\{x\}?$ It's a Borelian and we can't use the same argument as above. Or Am I understanding wrong the hint? $\endgroup$ – Squird37 Dec 31 '17 at 22:23

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