Regularity Properties of Baire Measures

Let $X$ be a separable, compact, Hausdorff space. Show that every closed set is a $G_{\delta }$ set.

Let $X$ be a Hausdorff space and $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a $\sigma$-finite Borel measure. Show that $\mu$ is Radon if and only if it is regular.

Show that a Hausdorff space $X$ is both locally compact and $\sigma$-compact if and only if there is an ascending countable collection $\left\{ \mathcal{O}_{k}\right\}_{k = 1}^{\infty }$ of open subsets of $X$ that covers $X$ and for each $k$,

Let $x_{0}$ be a point in the locally compact, Hausdorff space $X$. Is the Dirac delta measure concentrated at $x_{0}, \delta_{x_{0}}$, a regular Baire measure?

Let $X$ be an uncountable set with the discrete topology and $X^{*}$ its Alexandroff compactification with $x^{*}$ the point at infinity. Show that the singleton set $\left\{ x^{*}\right\}$ is a Borel set that is not a Baire set.

Let $X$ be a locally compact, Hausdorff space. Show that a Borel measure $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ is Radon if and only if every Borel set is measurable with respect to the Carathéodory measure induced by the premeasure $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$.

Prove Lemmas 23, 24, and 25.

Let $X$ be a compact, Hausdorff space and $f_{1}, \ldots , f_{n}$ continuous real-valued functions on $X$. Let $\nu$ be a signed Radon measure on $X$ with $\left|\nu \right|(X) \leq 1$ and let $c_{i} = \int_{X} f_{i} d \nu$, for $1 \leq i \leq n$.