Show that all complex Radon measures on a locally compact and $\sigma$-compact Hausdorff space is a Banach space Let $X$ be a locally compact Hausdorff space which is also $\sigma$-compact, and let $M(X)$ be the vector space of all complex Radon measures with the total variation norm $\|\mu\|:=|\mu|(X)$. Show that $M(X)$ is a Banach space.
I know that we can invoke the Riesz representation theorem to conclude that $M(X)\equiv C_0(X\to\mathbf{C})^*$, since the dual is always a Banach space, then the claim follows. I want to prove it without using the Riesz representation theorem. Let $(\mu_n)$ be a Cauchy sequence in $M(X)$, we observe that 
$$|\mu_m(E)-\mu_n(E)|\leq |\mu_m-\mu_n|(E)\leq \|\mu_m-\mu_n\|$$
for any measurable set $E$, thus $(\mu_n(E))$ is a Cauchy sequence, we can then define $\mu(E):=\lim_{n\to\infty}\mu_n(E)$. But I get stuck here. I don't know how to show the following two statments


*

*$\mu(E)$ is a complex Radon measure.

*$(\mu_n)$ converges to $\mu$ in $M(X)$.


For the first statement, let $E_1,E_2,\dots$ be disjoint measurable sets of $X$, we have to show
$$\mu(\bigcup_{m=1}^\infty E_m)=\sum_{m=1}^\infty \mu(E_m).$$
By the definition of $\mu$, this is equivalent to
$$\lim_{n\to\infty}\sum_{m=1}^\infty\mu_n(E_m)=\sum_{m=1}^\infty\lim_{n\to\infty}\mu_n(E_m).$$
 A: As is often the case, it is much easier to work with the following equivalent characterization of completeness of a normed vector space:

A normed vector space $(X, \|\cdot\|)$ is complete if and only if for each sequence $(x_n)_n$ in $X$ with $\sum_n \|x_n\| < \infty$, there is some $x \in X$ with $x = \sum_{n=1}^\infty x_n = \lim_{N\to\infty} \sum_{n=1}^N x_n$.

In your case, as you noted yourself, we know that $\mu(E) := \sum_{n=1}^\infty \mu_n (E)$ converges for every Borel set $E$.
Now, let $(E_n)_n$ be disjoint and let $\varepsilon >0$. There is $N$ with $\sum_{n=N}^\infty \|\mu_n\| < \varepsilon$. Hence,
$$
\bigg|\sum_\ell \mu(E_\ell) - \mu(\biguplus_\ell E_\ell)\bigg| \leq \bigg|\sum_{n=1}^N \bigg(\sum_\ell \mu_n (E_\ell) - \mu_n(\biguplus_\ell E_\ell)\bigg)\bigg| + \sum_{n=N}^\infty \bigg|\sum_\ell \mu_n (E_\ell) \bigg| + \sum_{n=N}^\infty \bigg|\mu_n (\biguplus_\ell E_\ell)\bigg| < 2\varepsilon,
$$
for every $\varepsilon > 0$.
Likwise, one can show regularity of $\mu$ and $\sum_{n=1}^N \mu_n \to \mu$ with convergence in $M(X)$.
A: If $X$ is a locally compact Hausdorff space, by Riesz Representation Theorem, $M(X)\cong C_0(X)^*$. Note that $C_0(X)$ is a normed vector space, then its dual space $M(X)$ is a Banach space.
Here, we do not require that $X$ is $\sigma$-compact.
