Functions, measures and distributions

## Functions, measures and distributions

Last updates: 10 April 2010

## Functions, measures and distributions

Let $G$ be a locally compact Hausdorff topological group and let $\mu$ be a Haar measure on $G.$ The support of a function $f$ is $suppf= g∈ G| f g ≠ 0 .$ If it exists, the convolution of functions ${f}_{1}:G\to ℂ$ and ${f}_{2}:G\to ℂ$ is the function $\left({f}_{1}*{f}_{2}\right):G\to ℂ$ given by $f 1 * f 2 g = ∫G f 1 h f 2 h -1 g d μ g .$ Define an involution on functions $f:G\to ℂ$ by Useful norms on functions $f:G\to ℂ$ are defined by $∥ f ∥ 1 = ∫G f g d μ g ,$ $∥ f ∥ 22 = ∫G f g 2 d μ g ,$ $∥f∥ ∞ =sup f g | g∈ G .$ If it exists, the inner product of functions ${f}_{1}:G\to ℂ$ and ${f}_{2}:G\to ℂ$ is $f 1 f 2 = ∫G f 1 g$ f 2 g -1 d μ g . The left and right actions of $G$ on functions $f:G\to ℂ$ are defined by Some spaces of functions are

Let $X$ be a topological space. A $\sigma$-algebra is a collection of subsets of $X$ which is closed under countable unions and intersections and contains the set $X.$ A Borel set is a set in the smallest $\sigma$-algebra $ℬ$ containing all open sets of $X.$ A Borel measure is a function $\mu :ℬ\to \left[0,\infty \right]$ which is countably additive, ie $μ ⊔ i=1 ∞ A i = ∑ i = 0 ∞ μ A i ,$ for every disjoint collection of ${A}_{i}$ from $ℬ.$ A regular Borel measure measure is a Borel measure which satisfies for all $E\in ℰ.$ A complex Borel measure is a function $\mu :ℬ\to ℂ$ which is countable additive. The total variation measure with respect to a complex Borel measure $\mu$ is the measure $\left|\mu \right|$ given by where the sup is over all countable collections $\left\{{E}_{i}\right\}$ of disjoint sets of $ℬ$ such that $∪ i E i =E .$ A regular complex Borel measure is a Borel measure on $X$ such that the total variation measure $\left|\mu \right|$ is regular. A measure $\lambda$ is absolutely continuous with respect to a measure $\mu$ if $\mu \left(E\right)=0$ implies $\lambda \left(E\right)=0.$

Let $\mu$ be a Haar measure on a locally compact group $G.$ Under the map $functions → measures f ↦ f g dμ g$ the group algebra $ℂG$ maps to measures $\nu$ with finite support, $l\left(G\right)$ maps to measures with countable support, and ${L}^{1}\left(G,\mu \right)$ maps to measures $\nu$ with countable support and ${L}^{1}\left(G,\mu \right)$ maps to measures which are absolutely continuous with respect to $\mu .$

Let $X$ be a locally compact Hausdorff topological space. Define Then ${C}_{c}\left(X\right)$ is a normed vector space (not always complete) under the norm $∥ f ∥ ∞ =sup f x | x∈ X .$ The completion ${C}_{0}\left(X\right)$ of $Cc\left(X\right)$ with respect to $\parallel •{\parallel }_{\infty }$ is a Banach space. A distribution is a bounded linear functional $\mu :{C}_{c}\left(X\right)\to ℂ.$ The Riesz representation theorem says that with the notation the regular complex Borel measures on $X$ are exactly the distributions on $X.$ The norm $\parallel \mu \parallel$ is the norm of $\mu$ as a linear functional $\mu :{C}_{c}\left(X\right)\to ℂ.$ Viewing $\mu$ as a measure, $\parallel \mu \parallel =\left|\mu \right|\left(X\right),$ where $\left|\mu \right|$ is the total variation measure of $\mu .$

The support of the distribution $\mu$ is the set of $x\in X$ such that for each neighbourhood $U$ of $x$ there is $f\in {C}_{c}\left(X\right)$ such that $\mathrm{supp}\left(f\right)\subseteq U$ and $\mu \left(f\right)\ne 0.$ Define If $\phi :X\to Y$ is a morphism of locally compact spaces then $φ * : ℰ c X → ℰ c Y is given by φ * μ f =μ f∘ φ ,$ for $f\in {C}_{c}\left(Y\right).$

Let $G$ be a locally compact topological group. Define an involution on distributions by $μ * f =μ f * , for f∈ C c G .$ The convolution of distributions is defined by $∫G f g d μ 1 * μ 2 g = ∫G ∫G f g 1 g 2 d μ 1 g 1 d μ 2 g 2 .$ The left and right actions of $G$ on distributions are given by

Let $X$ be a smooth manifold. The vector space ${C}^{\infty }\left(X\right)$ is a topological vector space under a suitable topology. A compactly supported distribution on $X$ is a continuous linear functional $\mu :{C}^{\infty }\to ℂ.$ Let and, for a compact subset $K\subseteq X,$ $ℰ 1 XK = μ ∈ ℰ 1 X | supp μ ⊆ K .$ If $\phi :X\to Y$ is a morphism of smooth manifolds then $φ * : ℰ 1 X → ℰ 1 Y is given by φ * μ f =μ f∘ φ .$