The Construction of Radon Measures

Let ($X, \mathcal{T}$) be a Hausdorff topological space. Show that $\mathcal{T}$ is a semi-ring if and only if $\mathcal{T}$ is the discrete topology.

Let $(X, \mathcal{T})$ be a topological space and $\mu : \mathcal{T} \rightarrow [0, \infty ]$ a premeasure. Assume that if $\mathcal{O}$ is open and $\mu (\mathcal{O}) < \infty$, then $\mu (\operatorname {bd} \mathcal{O}) = 0$. Show that every open set is $\mu^{*}$-measurable.

Show that the restriction of Lebesgue measure on the real line to the Borel $\sigma$-algebra is a Radon measure.

Show that the restriction of Lebesgue measure on the Euclidean space $\mathbb {R}^{n}$ to the Borel $\sigma$-algebra is a Radon measure.

Show that a Dirac delta measure on a topological space is a Radon measure.

Let $X$ be an uncountable set with the discrete topology and $\left\{ x_{k}\right\}_{1 \leq k < \infty }$ a countable subset of $X$. For $E \subseteq X$, define

Show that $2^{X} = \mathcal{B}(X)$ and $\mu : \mathcal{B}(X) \rightarrow$ is a Radon measure.

Show that the sum of two Radon measures also is Radon.

Let $\mu$ and $\nu$ be Borel measures on $\mathcal{B}(X)$, where $X$ is a compact topological space, and suppose that $\mu$ is absolutely continuous with respect to $\nu$. If $\nu$ is Radon show that $\mu$ also is Radon.

Let $(X, \mathcal{T})$ be a locally compact, Hausdorff space and $\mu : \mathcal{T} \rightarrow [0, \infty ]$ a premeasure for which the restriction to $\mathcal{B}(X)$ of $\mu^{*}$ is a Radon measure. Show that $\mu$ is a Radon premeasure.

Let $X$ be a locally compact, Hausdorff space and $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a Radon measure. Show that any Borel set $E$ of finite measure is inner regular in the sense that

Conclude that if $\mu$ is $\sigma$-finite, then every Borel set is inner regular.

Let $X$ be a topological space, $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a $\sigma$-finite Radon measure, and $E \subseteq X$ a Borel set. Show that there is a $G_{\delta }$ subset $A$ of $X$ and an $F_{\sigma }$ subset $B$ of $X$ for which

For a metric space $X$, show that $\mathcal{B}(X)$ is the smallest $\sigma$-algebra with respect to which all of the continuous real-valued functions on $X$ are measurable.