Let $X$ be a locally compact Hausdorff space.
Denote by $C(X)$ the space of continuous functions on $X$ and by $C_c(X) \subseteq C_0(X) \subseteq C_b(X) \subseteq C(X)$ the subspaces of continuous functions with compact support, vanishing at infinity and those that are bounded respectively.
- The $\sigma$-algebra $\sigma(C_c(X))$ generated by $C_c(X)$ coincides with the $\sigma$-algebra generated by all the compact $G_\delta$-sets [Aliprantis, Border, "Infinite dimensional Analysis", Lemma 4.64].
- The $\sigma$-algebra $\sigma(C_b(X)) = \sigma(C(X))$ coincides with $\sigma$-algebra generated by all the zero-sets, i.e. sets $f^{-1}(0)$ where $f \in C_b(X)$ or $f \in C(X)$.
Both $\sigma$-algebras are referred to as the Baire $\sigma$-algebra. But be aware that for a general locally compact Hausdorff space $X$, these $\sigma$-algebras can be different: $\sigma(C_c(X)) \subsetneq \sigma(C_b(X))$. [For instance, if $X$ is an uncountable discrete space then compact subsets are finite, hence $\sigma(C_c(X))$ is the countable-cocountable $\sigma$-algebra, while $\sigma(C_b(X))$ is the full power set $\sigma$-algebra.]
Just curious: Is there a similar characterization of $\sigma(C_0(X))$ in terms of a $\sigma$-algebra generated by a certain collection of subsets of $X$?
Best Answer
$C_0(X)$ generates the same $\sigma$-algebra as $C_c(X)$ (and thus the same $\sigma$-algebra as the compact $G_\delta$ sets). This follows from the fact that every element of $C_0(X)$ is the pointwise (in fact, uniform) limit of a sequence in $C_c(X)$: given $f\in C_0(X)$ and $\epsilon>0$, there is a compact set $K$ such that $|f|<\epsilon$ outside of $K$, and then we can extend $f|_K$ to a function with compact support that stays no larger than $\epsilon$ outside of $K$. Since a limit of a sequence of measurable functions is measurable, this means every function in $C_0(X)$ is $\sigma(C_c(X))$-measurable.