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=g∈G|fg≠0.
If it exists, the convolution
of functions f1:G→ℂ
and f2:G→ℂ is the function f1*f2:G→ℂ
given by f1*f2g=∫Gf1hf2h-1gdμg.
Define an involution on functions f:G→ℂ by f*g=fg-1,g∈G.
Useful norms on functions f:G→ℂ
are defined by ∥f∥1=∫Gfgdμg,
∥f∥ 2=∫Gfg2dμg,
∥f∥∞=supfg|g∈G.
If it exists, the inner product
of functions f1:G→ℂ and f2:G→ℂ is f1f2=∫Gf1gf2g-1dμg.
The left and right actions of G
on functions f:G→ℂ
are defined by Lgfx=fg-1x, and Rgfx=fxg,g,x∈G.
Some spaces of functions are ℂG=functions f:G→ℂ with finite support .
l1G=functions f:G→ℂ with countable support and f=∑g∈Gfg<∞.
L1Gμ=functions f:G→ℂ 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 ℬ
containing all open sets of X.
A Borel measure
is a function μ:ℬ→[0,∞]
which is countably additive, ie μ⊔i=1∞Ai=∑i=0∞μAi,
for every disjoint collection of Ai
from ℬ.
A regular
Borel measure measure is a Borel measure which satisfies μE=supμK|K⊆E, for K compact=infμU|U⊇E, for U open,
for all E∈ℰ.
A complex Borel measure is a function μ:ℬ→ℂ which is countable additive. The total variation
measure with respect to a complex Borel measure μ
is the measure μ
given by μE=sup∑iμEi,for E∈ℰ,
where the sup is over all countable collections Ei
of disjoint sets of ℬ
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 functions→measuresf↦fgdμg
the group algebra ℂG 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→ℂ.
Then CcX
is a normed vector space (not always complete) under the norm ∥f∥∞=supfx|x∈X.
The completion C0X
of CcX
with respect to ∥•∥∞
is a Banach space. A distribution
is a bounded linear functional μ:CcX→ℂ.
The Riesz representation theorem says that with the notation μf=∫Xfxdμx,f∈CcX,
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→ℂ.
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 x∈X
such that for each neighbourhood U
of x
there is f∈CcX
such that suppf⊆U
and μf≠0.
Define ℰcX=distributions μ on X with compact support .
If φ:X→Y
is a morphism of locally compact spaces then
φ*:ℰcX→ℰcYis given byφ*μf=μf∘φ,
for f∈CcY.
Let G
be a locally compact topological group. Define an involution on distributions by μ*f=μf*,forf∈CcG.
The convolution
of distributions is defined by ∫Gfgdμ1*μ2g=∫G∫Gfg1g2dμ1g1dμ2g2.
The left and right actions of G
on distributions are given by Lgμf=μLg-1,andRgμf=μRg-1f,for all f∈CcG.
Let X
be a smooth manifold. The vector space C∞X
is a topological vector space under a suitable topology. A compactly supported distribution on X
is a continuous linear functional μ:C∞→ℂ.
Let ℰ1X=μ:C∞→ℂ and, for a compact subset K⊆X,
ℰ1XK=μ∈ℰ1X|suppμ⊆K.
If φ:X→Y
is a morphism of smooth manifolds then φ*:ℰ1X→ℰ1Yis given byφ*μf=μf∘φ.
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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