Let $B$ be a Banach space of functions on a Radon space $X$. By the Hahn-Banach theorem, we know that the canonical evaluation map is isometric. That is, for every $f \in B$, we have $$\|f\| = \sup_{\varphi \in B^*, \,\|\varphi\|_* \leq 1} \varphi(f),$$ (and this supremum is achieved). Let $M(X)$ denote the set of signed Radon measures on $X$, some of which can be considered as elements of $B^*$ in the natural way. I would like to understand under which conditions $M(X) \cap B^*$ is dense in $B^*$ (with respect to the weak-* topology), so that $$\|f\| = \sup_{\varphi \in M(X) \cap B^*, \,\|\varphi\|_* \leq 1} \varphi(f).$$ This holds for many of the choices of $B$ I have considered. For example, if $B$ is the set of bounded measurable functions on $X$ equipped with the sup norm, then the norm of $B^*$ coincides with the total variation norm on $M(X)$. Then $\|f\|_\infty = \sup_{x \in X} \delta_x(f)$, where $\delta_x$ denotes the Dirac mass at $x \in X$ (and $\|\delta_x\|_* = 1$). As another example, suppose $(X,d)$ is a Polish metric space and $B$ is the space of Lipschitz functions on $X$, equipped with the Lipschitz norm $\|f\|_L = \sup_{x,y \in X}\frac{|f(x) - f(y)|}{d(x,y)}$ (with some centering, so that this is a proper norm). Then, we have $$\|f\|_L = \sup_{x,y \in X} \left(\frac{\delta_x - \delta_y}{d(x,y)}\right)(f),$$ and $\|\frac{\delta_x - \delta_y}{d(x,y)}\|_* = 1$ (with the dual norm being the Kantorovich-Rubinstein norm). Are there simple sufficient conditions on $X$ and $B$ so that this holds? The Riesz–Markov–Kakutani representation theorem seems potentially relevant here.
Edit 1: Here is one such condition. If $B$ is reflexive and $B^*$ contains the pointwise evaluation functionals (e.g. an RKHS on $X$), then the span of the Dirac deltas (and hence $M(X)$) is dense in $B^*$ (w.r.t. the weak topology), since the intersection of their kernels is trivial.
Edit 2: Actually, reflexivity is not needed! A subspace $V$ is dense in $B^*$ w.r.t. the weak-* topology if and only if $\varphi(f) = 0$ for every $\varphi \in V$ implies $f = 0$, which certainly holds for the Dirac deltas. This captures both of my examples (with appropriate centering for the Lipschitz case). $L^p$ spaces (for $p < \infty$ or a sigma-finite measure) satisfy my property (by the extremal version of Hölder's inequality), but are not captured by this.
