If is a locally compact Hausdorff topological group, with the algebra of compactly supported continuous functions from to , a Haar integral is a continuous linear map which is invariant under the aparent action of on . It turns out that there exists a unique such integral, up to a scalar multiple. The Riesz representation theorem then allows one to conclude the existence of a unique Haar measure, which is a -invariant Borel measure on .
The archetypal example of Haar measure is the Lebesgue measure on the (additive group underlying) cartesian space .
Let be a locally compact Hausdorff group. Let denote the vector space of continuous real-valued functionals with compact support on . This is a locally convex topological vector space where the locally convex structure is specified by the family of seminorms
ranging over compact subsets of . Recall that a Radon measure on may be described as a continuous linear functional
Such a Radon measure defines a measure on the -algebra of Borel sets in the usual sense of measure theory, where
In this context, a (left) Haar integral on is a nonzero such linear functional such that
for each and each , where sends to .
Correspondingly, a (left) Haar measure on is a nonzero Radon measure such that
for all and all Borel sets .
There is an analogy between Haar measure and scaled-cardinality on a finite group. In fact, the latter is a special case of the former, as we may view a finite group as a discrete topological group. While measure on a (discrete) finite group is subsumed by the notion of Haar measure, it may be of interest to build intuition.
Let be a finite group. Let be the category of -representations of -modules, where is a ring in which is invertible. This category is equivalent to -mod. We can view as a trivial -representation, where for each and each .
Let be a compactum. Let be the category of Banach representations of . Objects in are banach spaces over with a continuous norm preserving action , i.e. . Maps in are short maps which are -equivariant. (Alternatively, can be viewed as a category of certain -modules.) We can view as a trivial -representation, where for each and each .
Let be the ring of set-maps from to . acts on where sends to . There is a map of abelian groups sending to , analogous to the Haar integral. Indeed,
What corresponds to the Haar measure on is simply cardinality (though we must appropriately divide by the cardinality of , to get a function from the power set of to sending to ).
Using the Haar integral, we may define convolution product: sending to the map sending . This is analogous to the map sending to the map sending to .
In both cases, we get a “bar construction”. For the compact Hausdorff case, we get:
and for the finite case, we get:
Calling this a bar resolution is a slight abuse of terminology; the “monad” involved is actually has no unit, as has no unit for the convolution product. However, convolution makes an assocciative nonunitial algebra, so that the resolution is still a unitless monad. Hence there are evident face maps without degeneracies.
While has no unit, there is a canonical way of adding units to it: take products with . Write for the elements of ; since is a formal unit for convolution, we may think of it as a Dirac delta function. Now the bar resolution
has degeneracies as well.
Any locally compact Hausdorff topological group admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple. This result was first proven by Weil. A proof can be found in these online notes by Rubinstein-Salzedo. A different, constructive proof, due to E.M. Alfsen, can be found in the article “A simplified and constructive proof of the existence and uniqueness of haar measure”.
We here give a lesser known proof of the existence of the Haar integral, specifically on compact Hausdorff groups , which uses convex sets and the Krein Milman theorem instead of measure theory. Let be the category of Banach representations of (see “Analogy with the Finite Case”).
is such a Banach representation. We may view as a Banach representation of as well, where for each and each . embeds into as constant functions. We may then consider the exact sequence
A Haar integral on the -representation is equivalently a retract for the injection . In other words, it is a function such that
The last of these requirements, given the others, is equivalent to continuity of .
In some sense, we might wish to show that vanishes; this would show that the sequence
splits by the usual characterization of extensions via . On further contemplation, it is sufficient to show that the trivial -representation is an injective object in . This could be seen as an equivariant Hahn-Banach theorem.
Proof: We show that is an injective object in . Take an injection of Banach representations of , . Let be a map of Banach representations of . By the (usual) Hahn-Banach theorem, there exists a map in the category of Banach spaces and short maps extending , though it may lack -invariance.
Consider the subset of all extensions of to . Let be the collection of -invariant compact convex subsets of this set. contains the convex hull of , where is some chosen extension of to , so is nonempty. Using compactness and Zorn's Lemma, we may find a minimal element of in this collection, where is ordered where when . Call this element . must be a singleton. If contains a point which is not extremal then it contains the convex hull of the orbit of that point, which would be a proper -invariant compact convex subset of (see Krein Milman theorem). Therefore is a singleton, and its unique element is a -invariant functional extending .
In particular, since has been shown to be injective, the map lifts along the inclusion
giving a retract for in .
It follows that has norm , and from this positivity follows immediately.
In Lawvere’s thinking about extensive and intensive quantities,
is a space of intensive quantities on .
is the space of extensive quantities on , where is the category of Banach spaces with bounded maps as maps. The Haar integral is the unique -invariant element of this space of norm .
the integration map is the canonical evaluation pairing
If we suggestively write for , then becomes a way of writing . In particular, choosing to be the Haar measure, we can write as .
The left and the right Haar measure may or may not coincide, groups for which they coincide are called unimodular. Consider the matrix subgroup
The left and right invariant measures are, respectively,
and so G is not unimodular.
Abelian groups are obviously unimodular; so are compact groups and discrete groups.
Last revised on April 8, 2021 at 06:23:49. See the history of this page for a list of all contributions to it.