Functions, measures and distributions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

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 $$\mathrm{supp}f=\left\{g\in G|f\left(g\right)\ne 0\right\}.$$ 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 $$\left(f_1*f_2\right)\left(g\right)={\int }_G{f}_1\left(h\right)f_2\left(h^{-1}g\right)d\mu \left(g\right).$$ Define an involution on functions $f:G\to ℂ$ by $$f^{*}\left(g\right)=f\left(g^{-1}\right),\text{for all }g\in G.$$ Useful norms on functions $f:G\to ℂ$ are defined by $${\parallel f\parallel }_1={\int }_G\left|f\left(g\right)\right|d\mu \left(g\right),$$ $${\parallel }_f^{\parallel }22={\int }_G{\left|f\left(g\right)\right|}^{2}d\mu \left(g\right),$$ $${\parallel f\parallel }_{\infty }=\sup \left\{\left|f\left(g\right)\right||g\in G\right\}.$$ If it exists, the inner product of functions $f_1:G\to ℂ$ and $f_2:G\to ℂ$ is $$⟨f_1,f_2⟩={\int }_G{f}_1\left(g\right)$$ f 2 g -1 d μ g . The left and right actions of $G$ on functions $f:G\to ℂ$ are defined by $$\left(L_{g}f\right)\left(x\right)=f\left(g^{-1}x\right),\text{ and }\left(R_{g}f\right)\left(x\right)=f\left(xg\right),g,x\in G.$$ Some spaces of functions are $$ℂG=\left\{\text{functions }f:G\to ℂ\text{ with finite support }\right\}.$$ $$l^1\left(G\right)=\left\{\text{functions }f:G\to ℂ\text{ with countable support and }∥f∥=\sum _{g\in G}\left|f\left(g\right)\right|<\infty \right\}.$$ $$L^1\left(G,\mu \right)=\left\{\text{functions }f:G\to ℂ\text{ such that }∥f∥={\int }_G\left|f\left(g\right)\right|d\mu \left(g\right)<\infty \right\}.$$

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 $\mu :ℬ\to [0,\infty ]$ which is countably additive, ie $$\mu \left(\bigsqcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=0}^{\infty }\mu \left(A_i\right),$$ for every disjoint collection of $A_i$ from $ℬ.$ A regular Borel measure measure is a Borel measure which satisfies $$\mu \left(E\right)=\sup \left\{\mu \left(K\right)|K\subseteq E,\text{ for }K\text{ compact}\right\}=\inf \left\{\mu \left(U\right)|U\supseteq E,\text{ for }U\text{ open}\right\},$$ 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 $$\left|\mu \right|\left(E\right)=\sup \sum _i\left|\mu \left(E_i\right)\right|,\text{for }E\in ℰ,$$ where the sup is over all countable collections $\left\{E_i\right\}$ of disjoint sets of $ℬ$ such that $$\underset{i}{\cup }{E}_i=E.$$ 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 $$\begin{array}{rcl}\left\{\text{functions}\right\}& \to & \left\{\text{measures}\right\}\\ f& \mapsto & f\left(g\right)d\mu \left(g\right)\end{array}$$ 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 $$C_c\left(X\right)=\left\{\text{continuous functions }f:X\to ℂ\text{ with compact support}\right\}.$$ Then $C_c\left(X\right)$ is a normed vector space (not always complete) under the norm $$\parallel f{\parallel }_{\infty }=\sup \left\{\left|f\left(x\right)\right||x\in X\right\}.$$ 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 $$\mu \left(f\right)={\int }_{X}f\left(x\right)d\mu \left(x\right),\text{for all }f\in C_c\left(X\right),$$ 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 $\mathrm{supp}\mu$ 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 $${ℰ}_c\left(X\right)=\left\{\text{distributions }\mu \text{ on }X\text{ with compact support }\right\}.$$ If $\phi :X\to Y$ is a morphism of locally compact spaces then $${\phi }_{*}:{ℰ}_c\left(X\right)\to {ℰ}_c\left(Y\right)\text{is given by}\left({\phi }_{*}\mu \right)\left(f\right)=\mu \left(f\circ \phi \right),$$ for $f\in C_c\left(Y\right).$

Let $G$ be a locally compact topological group. Define an involution on distributions by $${\mu }^{*}\left(f\right)=\mu \left(f^{*}\right),\text{for}f\in C_c\left(G\right).$$ The convolution of distributions is defined by $${\int }_{G}f\left(g\right)d\left({\mu }_1*{\mu }_2\right)\left(g\right)={\int }_G{\int }_{G}f\left(g_1{g}_2\right)d{\mu }_1\left(g_1\right)d{\mu }_2\left(g_2\right).$$ The left and right actions of $G$ on distributions are given by $$\left(L_g\mu \right)\left(f\right)=\mu \left(L_{g^{-1}}\right),\text{and}\left(R_g\mu \right)\left(f\right)=\mu \left(R_{g^{-1}}f\right),\text{for all }f\in C_c\left(G\right).$$

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 $${ℰ}^1\left(X\right)=\left\{\text{continuous linear functionals }\mu :C^{\infty }\to ℂ\right\}$$ and, for a compact subset $K\subseteq X,$ $${ℰ}^1\left(X,K\right)=\left\{\mu \in {ℰ}^1\left(X\right)|\mathrm{supp}\left(\mu \right)\subseteq K\right\}.$$ If $\phi :X\to Y$ is a morphism of smooth manifolds then $${\phi }_{*}:{ℰ}^1\left(X\right)\to {ℰ}^1\left(Y\right)\text{is given by}\left({\phi }_{*}\mu \right)\left(f\right)=\mu \left(f\circ \phi \right).$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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