Here is the statement...

Suppose that $(X,\tau)$ is a comletely regular topological (I think the lecturer requires X to be Hausdorff too.), and that E is a dense linear subspace of $(C_{b}(X),\|\cdot\|_{\infty})$. Let $\sigma (E)$ be the smallest $\sigma$-field which makes all the functions in E measurable. Then $\sigma (E)$ is the Baire $\sigma$-field.

Why does E have to be a linear space?

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    $\begingroup$ Which makes all the functions on E measurable? Really? $\endgroup$ – Chris Eagle Mar 7 '13 at 16:57
  • $\begingroup$ @Chris Good spot. I meant in E. $\endgroup$ – user58514 Mar 7 '13 at 16:58
  • $\begingroup$ So $\sigma(E)$ is supposed to be a $\sigma$-field on $X$? $\endgroup$ – Chris Eagle Mar 7 '13 at 16:58
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    $\begingroup$ As Chris points out, the word "continuous" is missing. $\endgroup$ – Davide Giraudo Mar 7 '13 at 16:58
  • $\begingroup$ @Chris Yes. Sorry I didn't make that clear! $\endgroup$ – user58514 Mar 7 '13 at 16:59

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