Complex Measures: Lebesgue

Question

Given a Borel space $\Omega$.

Consider a complex measure: $$\mu:\mathcal{B}(\Omega)\to\mathbb{C}$$

Regard a sequence: $$\eta_n\in\mathcal{L}(\Omega):\quad\eta_n\to\eta$$

Suppose one finds: $$\overline{\eta}\in\mathcal{L}(\mu):\quad|\eta_n|\leq|\overline{\eta}|$$

Then it follows: $$\int\eta\,\mathrm{d}\mu=\lim_n\int\eta_n\mathrm{d}\mu$$

How can I check this?

Reference

This is a lemma for Pushforward (SM)

Answer 1

For the density it is: $$|\eta_n|\cdot|u|=|\eta_n|\cdot1\leq|\overline{\eta}|$$

So one obtains: $$\lim_n\int\eta_n\,\mathrm{d}\mu=\lim_n\int\eta_n\cdot u\mathrm{d}|\mu|=\int\eta\cdot u\mathrm{d}|\mu|=\int\eta\,\mathrm{d}\mu$$

Concluding the assertion.