Complex Haar L^p spaces and compactly supported functions

Let $X$ be an LCH space with a fixed Radon measure $\mu$, and write $C_c(X;\mathbb{C})$ for the space of continuous complex-valued functions of compact support (Compact support, $C_c(X)$, and $C_0(X)$). A complex-valued function on $X$ is measurable when its real and imaginary parts are measurable, and it is integrable when its modulus is integrable (Integrable real and complex functions, and their integrals). For $p\in \{1,2\}$ the space $L^p(X,\mu ;\mathbb{C})$ consists of the almost-everywhere equivalence classes of measurable complex functions $f$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_p:=({\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu {)}^{1\mathrm{/}p}<\mathrm{\infty },$ the quotient and class notation being those of The space $L^p(\mu )$ as the quotient by null functions. The modulus $\mathrm{\mid }f\mathrm{\mid }$ of a complex measurable function is measurable and depends only on the class of $f$ up to a null set, so $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$ is well defined on classes; on classes it is the norm of a complex normed space, and $L^1(X,\mu ;\mathbb{C})$ and $L^2(X,\mu ;\mathbb{C})$ are the complex Lebesgue spaces used on this page.

Let now $G$ be an LCH group with a fixed left Haar measure $\mu$ (Left Haar integral and left Haar measure). Then $C_c(G):=C_c(G;\mathbb{C})$ and $L^p(G):=L^p(G,\mu ;\mathbb{C})(p\in \{1,2\}),$ suppressing the measure from the notation; the Haar measure is kept fixed throughout the page, so no ambiguity arises. The subscript is written $\mathrm{\parallel }f{\mathrm{\parallel }}_1$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_2$ for the two norms.

• Every $C_c$ function lies in both spaces. If $f\in C_c(G)$ then $\mathrm{\mid }f\mathrm{\mid }$ is bounded on the compact set $\mathrm{supp}f$ and this set has finite $\mu$-measure, so $\int \mathrm{\mid }f\mathrm{\mid }d\mu <\mathrm{\infty }$ and $\int \mathrm{\mid }f{\mathrm{\mid }}^{2}d\mu <\mathrm{\infty }$; the class of $f$ in $L^p(G)$ is therefore defined for $p=1,2$.
• Why the complex spaces are defined locally. The quotient construction of The space $L^p(\mu )$ as the quotient by null functions is stated for real-valued functions and its norm theory is developed there for that case; the present definition fixes the complex version and its notation before convolution and the regular representations use it. No additional choice is needed here once $\mu$ is fixed.
• Finiteness of the Haar measure on compact sets. The integrability just claimed uses that a left Haar measure is finite on compact sets, which is part of the definition of a Radon measure recorded in Left Haar integral and left Haar measure.