Let
- $\Omega$ be a metric space,
- $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
- $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$.
For $\mu \in \mathcal{M}(\Omega)$, we denote by $|\mu|$ its associated variation measure. We say that a sequence $\left\{\mu_n\right\} \subset \mathcal{M}(\Omega)$ converges to $\mu \in \mathcal{M}(\Omega)$ weakly if $\int_\Omega f \mathrm d \mu_n \to \int_\Omega f \mathrm d \mu$ for all $f \in C_b(\Omega)$ and we write $\underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu_n= \mu$;
It is proved in this answer that
Theorem Let $\mu_n,\mu\in \mathcal{M}(\Omega)$ such that that $\underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu_n=\mu$. Then for any open subset $\Theta$ of $\Omega$, $$ |\mu|(\Theta) \leq \liminf _{n \rightarrow \infty}\left|\mu_n\right|(\Theta) . $$
The essential part of the proof is that given $\varepsilon>0$ there is $f \in C_b(\Omega)$ such that $$ |f| \le 1_\Theta \quad \text{and} \quad \int_\Omega f \mathrm d\mu\ge|\mu|(\Theta)-\varepsilon. $$
Then by weak convergence of $(\mu_n)$ we have $$ |\mu|(\Theta)-\varepsilon \leq \int f \mathrm{~d} \mu=\lim _{n \rightarrow \infty} \int f \mathrm{~d} \mu_n \leq \liminf _{n \rightarrow \infty} \int|f| \mathrm{d}\left|\mu_n\right| \leq \liminf _{n \rightarrow \infty}\left|\mu_n\right|(\Theta). $$
The result then follows by taking the limit $\varepsilon \to 0^+$.
My understanding: To have above inequalities, we use the fact that $\mu,\mu_n$ are real-valued.
Can above theorem be extended to complex Borel measures?
Question: Can the above theorem be extended to the setting of complex Borel measures?
Update: Below is my failed attempt. It would be great if it can be fixed into a valid proof. I could not prove that $$ \liminf _{n \rightarrow \infty} \big (\left|\mu^1_n\right|(\Theta) + \left|\mu^2_n\right|(\Theta) \big ) \le \liminf_n \left|\mu_n\right|(\Theta) . $$
My attempt: Let $\mu_n, \mu$ are complex Borel measures on $\Omega$ such that $\underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu_n= \mu$. Assume $\mu_n, \mu$ are decomposed into $\mu_n =\mu_n^1 + i \mu_n^2$ and $\mu =\mu^1 + i \mu^2$ where $i$ is the imaginary unit and $\mu_n^1, \mu_n^2, \mu^1, \mu^2$ are finite signed Borel measures. We have $$ \begin{align*} \underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu_n = \mu &\iff \int_\Omega f \mathrm d \mu_n^1 + i \int_\Omega f \mathrm d \mu_n^2 \to \int_\Omega f \mathrm d \mu^1 + i \int_\Omega f \mathrm d \mu^2 \quad \forall f \in C_b(\Omega) \\ &\iff \int_\Omega f \mathrm d \mu_n^1 \to \int_\Omega f \mathrm d \mu^1 \quad \text{and} \quad \int_\Omega f \mathrm d \mu_n^2 \to \int_\Omega f \mathrm d \mu^2 \quad \forall f \in C_b(\Omega) \\ &\iff \underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu^1_n = \mu^1 \quad \text{and} \quad \underset{n \rightarrow \infty}{\operatorname{w-lim}} \, \mu^2_n = \mu^2 \\ &\implies |\mu^1|(\Theta) \leq \liminf _{n \rightarrow \infty}\left|\mu^1_n\right|(\Theta) \quad \text{and} \quad |\mu^2|(\Theta) \leq \liminf _{n \rightarrow \infty}\left|\mu^2_n\right|(\Theta) \end{align*} $$ for all open subsets $\Theta$ of $\Omega$. From this question, we have $$ |\mu|(\Theta) \le |\mu^1|(\Theta) + |\mu^2|(\Theta). $$
As such, $$ |\mu|(\Theta) \le \liminf _{n \rightarrow \infty}\left|\mu^1_n\right|(\Theta) + \liminf _{n \rightarrow \infty}\left|\mu^2_n\right|(\Theta) \le \liminf _{n \rightarrow \infty} \big (\left|\mu^1_n\right|(\Theta) + \left|\mu^2_n\right|(\Theta) \big ). $$
