Let $(\Omega,\mathscr{A})$ be a measurable space and $\mu\colon\mathscr{A}\to\mathbb{C}$ a complex measure (i.e. a $\sigma$-additive function from $\mathscr{A}$ to $\mathbb{C}$). Then $\mu$ can be decomposed into a (unique) real and imaginary part. In other words, there are (finite) signed measures $\mu_{1},\mu_{i}\colon\mathscr{A}\to\mathbb{R}$ such that $\mu=\mu_{1}+i\mu_{i}$. The Jordan decomposition lets us split the real and imaginary part of $\mu$ into finite positive measures $\mu_{1}^{+},\mu_{1}^{-},\mu_{i}^{+},\mu_{i}^{-}\colon\mathscr{A}\to\mathbb{R}_{\geq0}$ such that $\mu_{1}=\mu_{1}^{+}-\mu_{1}^{-}$ and $\mu_{i}=\mu_{i}^{+}-\mu_{i}^{-}$. So we have $$\mu=(\mu_{1}^{+}-\mu_{1}^{-})+i(\mu_{i}^{+}-\mu_{i}^{-}).$$ Then a measurable complex valued function $f\colon\Omega\to\mathbb{C}$ is called integrable with respect to the complex measure $\mu$ if and only if the real and imaginary part of $f$ are both integrable with respect to the measures $\mu_{1}^{+}$, $\mu_{1}^{-}$, $\mu_{i}^{+}$ and $\mu_{i}^{-}$. If $f=f_{1}+if_{i}$ (with $f_{1}$ and $f_{i}$ real-valued) is integrable with respect to $\mu$, then its integral is defined to be \begin{align*}\int_{\Omega}f \ \text{d}\mu&:=\int_{\Omega}f_{1} \ \text{d}\mu_{1}^{+}-\int_{\Omega}f_{1} \ \text{d}\mu_{1}^{-}+i\int_{\Omega}f_{1} \ \text{d}\mu_{i}^{+}-i\int_{\Omega}f_{1} \ \text{d}\mu_{i}^{-}\\ &=\int_{\Omega}f_{i} \ \text{d}\mu_{1}^{+}-\int_{\Omega}f_{i} \ \text{d}\mu_{1}^{-}+i\int_{\Omega}f_{i} \ \text{d}\mu_{i}^{+}-i\int_{\Omega}f_{i} \ \text{d}\mu_{i}^{-}. \end{align*} I was studying complex measures, but the concept of $L^{p}$ spaces is very unclear to me in this case. I can't find any proper literature about it either. Can someone explain to me what is going on here?
I have a few questions in particular:
- Can we make sense of the space $L^{p}(\Omega,\mathscr{A},\mu)$ for $1\leq p\leq\infty$?
- Can we define a $p$-norm on this space? Is this norm also complete?
- In the case $p=2$, do we still have a (complete) inner-product space?
I think the total variation $|\mu|$ (which is a finite positive measure) of a complex measure $\mu$ has something to do with it, but I can't figure it out the details myself.
Any help is greatly appreciated!