# Completely Regular Topological Space and Measure Theory

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?

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