Proposition 12.4.2.label Let $E$ be a separable normed vector space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets:
- (1)
Open sets in $E$ with respect to the strong topology.
- (2)
$\bracs{B(x, r)|x \in E, r > 0}$.
- (3)
$\bracsn{\ol{B(x, r)}|x \in E, r > 0}$.
- (4)
Open sets in $E$ with respect to the weak topology.
Proof. (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3): By Proposition 18.2.5.
(4) $\subset$ (1): Every weakly open set is strongly open.
(2) $\subset$ (4): By Proposition 11.8.7, $\norm{\cdot}_{E}: E \to [0, \infty)$ is Borel measurable with respect to the weak topology. For any $x \in E$, let
then $\phi_{x}$ is Borel measurable with respect to the weak topology, so $B(x, r) = \bracs{\phi_x < r}$ is a Borel set with respect to the weak topology.$\square$