Interior of the cone of nonnegative Radon measures inside the space of Radon measures

Question

Let $X$ be a "sufficiently nice" space (say locally compact Hausdorff). We consider the space $\mathcal{M}(X)$ of finite signed Radon measures on $X$, as well as $\mathcal{M}^+(X)$, the cone of nonnegative measures inside $\mathcal{M}(X)$.

There are a number of standard topologies one can place on $\mathcal{M}(X)$ (such as: the norm topology coming from the total variation of measures, the weak* topology coming from the duality with the space $C_0(X)$ of continuous functions converging to zero towards infinity, etc.). For which topologies does $\mathcal{M}^+(X)$ have a non-empty interior inside $\mathcal{M}(X)$?

Answer 1

If $\mathcal M(X)$ is infinite dimensional, the non-negative cone has empty interior w.r.t. the norm topology:

Let $\mu \in \mathcal M^+(X)$ and $\varepsilon > 0$ be given. Then, there exists $x \in X$ with $0 \le \mu(\{x\}) \le \varepsilon$:

• If $X$ is uncountable, there exists $x \in X$ with $\mu(\{x\}) = 0$.
• If $X$ is countable, the sum $\sum_{x \in X} \mu(\{x\})$ converges, thus we can find $x \in X$ with $\mu(\{x\}) \le \varepsilon$.

Now, the measue $\mu - 2 \varepsilon \delta_x$ is no longer non-negative, where $\delta_x$ is the Dirac measure sitting at $x$.