Lp classes and explicit representatives

Let $(X,\mathrm{\Sigma },\mu )$ be a measure space in Measure spaces. Give $\mathbb{R}$ the order topology ${\mathcal{T}}_{<}$ of The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua, which that supplier identifies with the usual real topology, and let $\mathcal{B}(\mathbb{R}):={\sigma }_{\mathbb{R}}({\mathcal{T}}_{<})$ be its Borel sigma-algebra as in The Borel sigma-algebra of a topological space. A measurable real function is a map $f:X\to \mathbb{R}$ measurable from $(X,\mathrm{\Sigma })$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ in the sense of A measurable function between measurable spaces. For $1\le p<\mathrm{\infty }$, let the representative space be ${\mathcal{L}}^p(\mu )=\{f:X\to \mathbb{R}:f\text{ is measurable and }{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu <\mathrm{\infty }\}$, exactly the supplier definition The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$. For $p=\mathrm{\infty }$, the representatives are the measurable real functions for which $\mu (\{x:\mathrm{\mid }f(x)\mathrm{\mid }>C\})=0$ for some finite $C\ge 0$. The level sets used here are measurable by Closure properties of measurable functions used by the integral. This finite-bound condition is exactly $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$ in The essential supremum of a measurable function with respect to a measure and The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions: if such a $C$ exists then the essential supremum is at most $C$; conversely, if the essential supremum is finite, The essential supremum is attained as the least essential bound proves that this finite value itself is an essential bound.

For measurable $f,g:X\to \mathbb{R}$, the difference $f-g$ is measurable by the addition and scalar-multiplication clauses of Closure properties of measurable functions used by the integral. Therefore

$\{x:f(x)\mathrm{\ne }g(x)\}=(f-g{)}^{-1}(\mathbb{R}\setminus \{0\})\in \mathrm{\Sigma },$

because $\mathbb{R}\setminus \{0\}$ is open in the stated topology and hence Borel. On either representative space put

$f\sim g⟺\mu (\{x\in X:f(x)\mathrm{\ne }g(x)\})=0.$

By The null subspace of measurable functions that vanish almost everywhere and Measure-null sets and almost-everywhere statements relative to a measure, this is equivalent to $f-g\in \mathcal{N}(\mu )$, so it is exactly the relation in The space $L^p(\mu )$ as the quotient by null functions. The element $[f]\in L^p(\mu )$ is the equivalence class of $f$; an explicit representative is a specified measurable function belonging to that class. The quotient norms are

$\mathrm{\parallel }[f]{\mathrm{\parallel }}_p={\left({\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu \right)}^{1\mathrm{/}p}(1\le p<\mathrm{\infty }),\mathrm{\parallel }[f]{\mathrm{\parallel }}_{\mathrm{\infty }}={\mathrm{ess\hspace{0.17em}sup}}_X\mathrm{\mid }f\mathrm{\mid },$

with independence of representative and the norm axioms supplied by The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$.

A pointwise statement such as $f(x)=0$ concerns the selected representative. A property invariant under almost-everywhere equality, such as $f=0$ almost everywhere, concerns the class. No value of $[f]$ at a particular point is defined by this quotient.

This definition uses real-valued representatives. When a coupled Sobolev construction uses complex functions, it specifies their real and imaginary representatives separately; this item does not assert a complex-valued $L^p$ quotient or choose representatives simultaneously for an arbitrary family of classes. The definition imposes no additional choice principle.