We are given a real Fréchet space $\mathfrak X$. Let $\mathcal B(\mathfrak X)$ denote its Borel sigma algebra and $\mathfrak X'$ its topological dual. Consider the following set-function: $$ \mu: \mathcal B(\mathfrak X) \to \mathbb R $$ such that
i) $\mu(\emptyset)=0$,
ii) finitely additive,
iii) finite and regular,
i.e. $\mu$ is an element of $rba(\mathfrak X)$, if we use the classical notation.
Suppose that the following equation holds true $$ \mu[x\in\mathfrak X: \ell(x) < t]=0 $$ for any $\ell\in\mathfrak X'$ and $-\infty < t < \infty$.
Can one conclude that $\mu=0$?