Functions, measures and distributions

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 μ be a Haar measure on G. The support of a function f is suppf=gG|fg0. If it exists, the convolution of functions f1:G and f2:G is the function f1*f2:G given by f1*f2g=Gf1hf2h-1gdμg. Define an involution on functions f:G by f*g=fg-1,gG. Useful norms on functions f:G are defined by f1=Gfgdμg, f 2=Gfg2dμg, f=supfg|gG. If it exists, the inner product of functions f1:G and f2:G is f1f2=Gf1gf2g-1dμg. The left and right actions of G on functions f:G are defined by Lgfx=fg-1x,  and  Rgfx=fxg,g,xG. Some spaces of functions are G=functions  f:G with finite support  . l1G=functions  f:G with countable support and  f=gGfg<. L1Gμ=functions  f:G such that  f=Gfgdμg< .

Let X be a topological space. A σ-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 σ-algebra containing all open sets of X. A Borel measure is a function μ:[0,] which is countably additive, ie μi=1Ai=i=0μAi, for every disjoint collection of Ai from . A regular Borel measure measure is a Borel measure which satisfies μE=supμK|KE, for  K compact=infμU|UE, for  U open, for all E. A complex Borel measure is a function μ: which is countable additive. The total variation measure with respect to a complex Borel measure μ is the measure μ given by μE=supiμEi,for  E, where the sup is over all countable collections Ei of disjoint sets of such that iEi=E. A regular complex Borel measure is a Borel measure on X such that the total variation measure μ is regular. A measure λ is absolutely continuous with respect to a measure μ if μE=0 implies λE=0.

Let μ be a Haar measure on a locally compact group G. Under the map functionsmeasuresffgdμg the group algebra G maps to measures ν with finite support, lG maps to measures with countable support, and L1Gμ maps to measures ν with countable support and L1Gμ maps to measures which are absolutely continuous with respect to μ.

Let X be a locally compact Hausdorff topological space. Define CcX=f:X. Then CcX is a normed vector space (not always complete) under the norm f=supfx|xX. The completion C0X of CcX with respect to is a Banach space. A distribution is a bounded linear functional μ:CcX. The Riesz representation theorem says that with the notation μf=Xfxdμx,fCcX, the regular complex Borel measures on X are exactly the distributions on X. The norm μ is the norm of μ as a linear functional μ:CcX. Viewing μ as a measure, μ=μX, where μ is the total variation measure of μ.

The support supp μ of the distribution μ is the set of xX such that for each neighbourhood U of x there is fCcX such that suppfU and μf0. Define cX=distributions  μ on   X with compact support . If φ:XY is a morphism of locally compact spaces then φ*:cXcYis given byφ*μf=μfφ, for fCcY.

Let G be a locally compact topological group. Define an involution on distributions by μ*f=μf*,forfCcG. The convolution of distributions is defined by Gfgdμ1*μ2g=GGfg1g2dμ1g1dμ2g2. The left and right actions of G on distributions are given by Lgμf=μLg-1,andRgμf=μRg-1f,for all  fCcG.

Let X be a smooth manifold. The vector space CX is a topological vector space under a suitable topology. A compactly supported distribution on X is a continuous linear functional μ:C. Let 1X=μ:C and, for a compact subset KX, 1XK=μ1X|suppμK. If φ:XY is a morphism of smooth manifolds then φ*:1X1Yis given byφ*μf=μfφ.

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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