Is the set of probability measures on $[0,1]$ that induces a continuous distribution function weak * compact?

Question

I am an student in economics and I am trying to solve a fixed point problem, where the inputs of the function are probability measures. I'm trying to figure out whether the below results are true, but I have no clue by the best of my knowledge.

Let $\Delta([0,1])$ be the set of probability measures on the space $([0,1], \mathcal{B}[0,1])$, endowed with the weak-$*$ topology. Since $[0,1]$ is compact and Polish space, I know that $\Delta([0,1])$ is also compact and Polish with the weak$-*$ topology, metrizable with the Levy-Prokhorov metric.

Moreover, each probability measure $\mu \in \Delta([0,1])$ induces a distribution function $F_{\mu}: [0,1]\rightarrow [0,1]$ given as $F_{\mu}(x)=\mu([0,x])$.

By definition, any $F_{\mu}$ is weakly increasing and right-continuous. However, in my problem, I want to focus only on measures that induce a continuous distribution function.

Therefore, let us define $$\Delta^C([0,1])=\{\mu \in \Delta^C([0,1])| F_{\mu} \text{ is continuous} \}$$

My question is:

1. Is $\Delta^C([0,1])$ also compact in the weak$*$ topology?

I know that the Helly space is compact with respect to the product topology, but I am not sure with sup-norm or pointwise convergence topology.

2. If we define $A = \{ f| f\in [0,1]^{[0,1]} |\text{f is right continuous and weakly increasing }\}$, then is the function $F_{\cdot}: \Delta[0,1]\rightarrow A$ mapping $\mu$ into $F_{\mu}$ surjective/bijective?

Answer 1

Consider the probability measures $\mu_n$ that are uniform on $[0,1/n]$ for positive integers $n$. These are in $\Delta^C([0,1])$, but their weak-* limit is the unit point mass at $0$, which is not.