> [!definition] > > Let $X$ be a [[Locally Compact Hausdorff Space|LCH]] space, and denote $M(X)$ as the set of all complex [[Radon Measure|Radon measures]] on $X$. For any $\mu \in M(X)$, define > $ > \norm{\mu} = \abs{\mu}(X) > $ > where $\abs \mu$ is the total variation. Then *any* complex Borel measure on $X$ is Radon if and only if $\abs{\mu}$ is Radon, and $M(X)$ forms a [[Normed Vector Space|normed space]]. > > *Proof*. First note that any finite positive Borel measure is Radon if and only if it is regular on all Borel sets. Applying bounds on the real/complex and [[Jordan Decomposition Theorem|Jordan decomposition]] yields the equivalence. > > The norm property comes from taking [[Lebesgue-Radon-Nikodym Theorem|Radon-Nikodym derivatives]]. > > By the [[Riesz Representation Theorem]], this space is always complete.