${\mathcal{L}}^p$ and $L^{\mathrm{\infty }}$ are vector spaces for $p\ge 1$

Fix $1\le p<\mathrm{\infty }$ and let $f,g\in {\mathcal{L}}^p(\mu )$ and $a\in \mathbb{R}$. Measurability of $f+g$ and $af$ follows from [L2]. Pointwise, $\mathrm{\mid }f+g{\mathrm{\mid }}^p\le (2\max \{\mathrm{\mid }f\mathrm{\mid },\mathrm{\mid }g\mathrm{\mid }\}{)}^p\le 2^p(\mathrm{\mid }f{\mathrm{\mid }}^p+\mathrm{\mid }g{\mathrm{\mid }}^p)\mathrm{.}$ By [L3], $\int \mathrm{\mid }f+g{\mathrm{\mid }}^{p}d\mu \le 2^p(\int \mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu +\int \mathrm{\mid }g{\mathrm{\mid }}^{p}d\mu )<\mathrm{\infty }\mathrm{.}$ If $a\mathrm{\ne }0$, positive homogeneity in [L3] gives $\int \mathrm{\mid }af{\mathrm{\mid }}^{p}d\mu =\mathrm{\mid }a{\mathrm{\mid }}^p\int \mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu <\mathrm{\infty }$; if $a=0$, the integral is zero without forming $0\cdot \mathrm{\infty }$. Thus $f+g,af\in {\mathcal{L}}^p(\mu )$. The zero function belongs to the class, and additive inverses use $a=-1$.

Let $f,g\in L^{\mathrm{\infty }}(\mu )$ with $M:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $N:=\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$. There are measurable null sets $E_f,E_g$ such that $\mathrm{\mid }f\mathrm{\mid }\le M$ on $X\setminus E_f$ and $\mathrm{\mid }g\mathrm{\mid }\le N$ on $X\setminus E_g$. With $E:=E_f\cup E_g$, $E$ is measurable and null, and on $X\setminus E$ one has [L1, L2, L4, L5, given] $\mathrm{\mid }f+g\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }+\mathrm{\mid }g\mathrm{\mid }\le M+N,\mathrm{\mid }af\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }M\mathrm{.}$ Thus $f+g$ and $af$ are essentially bounded; measurability again comes from [L2].

The pointwise addition and scalar-multiplication identities are inherited from real-valued functions. Hence [L6] makes ${\mathcal{L}}^p(\mu )$ a real vector space for every $1\le p<\mathrm{\infty }$.

The pointwise identities are again inherited from real-valued functions, so [L6] makes $L^{\mathrm{\infty }}(\mu )$ a real vector space.