Proof of The Finite Signed Borel Measures on a Locally Compact Separable Metric Space Form a Banach Space under the Total Variation Norm

Throughout, $\Phi$ is the function of Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (i), and distances in $M(X)$ and in $C_{c}(X)^{*}$ are those of the norm metric.

By Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (iv), $\Phi$ is an isometric linear bijection of $M(X)$ onto $C_{c}(X)^{*}$.

Cauchy sequences converge. Let $(\nu_{m})$ be a Cauchy sequence in $M(X)$. By the isometry, $(\Phi(\nu_{m}))$ is a Cauchy sequence in $C_{c}(X)^{*}$. Since $(C_{c}(X),\|\cdot\|_{u})$ is a real normed space, its dual $C_{c}(X)^{*}$ with the dual norm is a real Banach space by The Dual of a Real Normed Space Is a Real Banach Space; Weak-Star Limits (iii), so $(\Phi(\nu_{m}))$ converges to some $\Lambda\in C_{c}(X)^{*}$ (Real Banach Space (i), Complete Metric Space (i)). Since $\Phi$ is onto, $\Lambda=\Phi(\nu)$ for some $\nu\in M(X)$, and by the isometry

$$\|\nu_{m}-\nu\|=\|\Phi(\nu_{m})-\Phi(\nu)\|_{*}\qquad(m\in\mathbb{N}).$$

Hence for every $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\|\nu_{m}-\nu\|<\varepsilon$ for every $m\ge N$, that is, $(\nu_{m})$ converges to $\nu$ in $M(X)$. So $M(X)$ is complete under the norm metric of $\|\cdot\|$, which proves (i).