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Home , Haar measure, Locally compact group

The Banach–Tarski Paradox and Amenability Lecture 9: Haar

25 August 2011 Topological groups Definition A topological is a group G that is also a Hausdorff topological space such that the group operations (g, h) 7→ gh and g 7→ g −1, for all g, h ∈ G, are continuous functions. Examples

× 1. The multiplicative group R of non-zero real numbers is a , topologised as an open subset of R. n2 2. Identify the set Mn,n(R) of n × n real matrices with R with its usual topology. Since det : Mn,n(R) → R is a continuous map, the group

GL(n, R) = {g ∈ Mn,n(R) : det(g) 6= 0}

n2 can be identified with an open subset of R . With the inherited topology, matrix multiplication and taking inverses are continuous operations, so GL(n, R) is a topological group. Locally compact groups

Definition A topological space X is locally compact if for each x ∈ X , there is a compact set K and an open set U such that x ∈ U ⊆ K. Definition A locally compact topological group or locally is a topological group G which is locally compact as a topological space. Examples

× 1. R is a . 2. GL(n, R) is a locally compact group, since every open subset n2 of R is locally compact in the inherited topology.

Definition Let G be a locally compact group. A (left-invariant) Haar measure on G is a regular such that for all g ∈ G and all Borel sets A, µ(gA) = µ(A). Theorem (Haar, Weil 1930s) Let G be a locally compact group. Then G has a left-invariant Haar measure. A left-invariant Haar measure is unique up to scalar multiples. Topological manifolds

Let X be a topological space. We say that X is an n–dimensional topological manifold if for every x ∈ X there is an open set U ⊂ X containing x and a homeomorphism from U onto an open subset n V of R . Examples

× 1. R is a 1–dimensional topological manifold. 2 2. GL(n, R) is a topological manifold of dimension n , since it is n2 an open subset of R . Smooth manifolds n n Let V be an open subset of R . A function f : V → R is smooth if it has continuous partial derivatives of all orders.

An n–dimensional topological manifold X is a smooth n–manifold if the following condition holds. Suppose x, x0 ∈ X have open neighbourhoods U, U0 such that there are homeomorphisms ϕ : U → V and ϕ0 : U0 → V 0, respectively, where V and V 0 are n open subsets of R . Then the function

0 −1 0 0 0 ϕ ◦ ϕ |ϕ(U∩U0) : ϕ(U ∩ U ) → ϕ (U ∩ U )

is smooth. Examples

× 1. R is a smooth 1–dimensional manifold. 2 2. GL(n, R) is a smooth n –manifold. The smoothness is because the map is polynomial. Change of variables in integration over R

In school you learnt that when integrating suitable functions f , putting x = g(u) for some suitable g : R → R the substitution rule gives

Z g(b) Z b Z b dx f (x)dx = f (g(u))g 0(u) du = f (g(u)) du g(a) a a du Haar measure on R×

× For any Borel subset A of R , we define µ(A) to be the Lebesgue Z 1 µ(A) := dx A |x| Then from properties of the integral, µ is countably additive, regular and non-zero. To see that µ is left-invariant, it’s enough to check that µ(A) = µ(λA) for all λ > 0 and A = (a, b). Using the substitution x = λu Z λb 1 Z b 1 Z b 1 µ(λA) = dx = λ du = du = µ(A) λa |x| a |λu| a |u| Similarly µ is right-invariant. × Therefore µ is a (left- and right-invariant) Haar measure on R . Change of variables in integration over Rn

In multivariable you learnt that when integrating suitable 2 2 functions f : R → R over A ⊂ R , putting x = g(u, v) and 2 y = h(u, v) for some suitable g, h : R → R ZZ ZZ f (x, y) dx dy = f (g(u, v), h(u, v)) Jac(T )du dv T (A) A

where  ∂x ∂x  ∂u ∂v Jac(T ) = det ∂y ∂y ∂u ∂v 2 2 is the Jacobian of the transformation T : R → R given by T (u, v) = (x, y) = (g(u, v), h(u, v)). n Similarly for maps f : R → R. Haar measure on GL(n, R) We will sketch the case n = 2 for the

+ GL(2, R) = {g ∈ GL(n, R) : det(g) > 0}

+ 4 Identify GL(2, R) with an open subset of R via   x11 x12 ←→ (x11, x12, x21, x22) x21 x22

+ For any Borel subset A of GL(2, R ) we define µ(A) to be the Lebesgue integral Z 1 µ(A) := 2 dx11dx12dx21dx22 A (x11x22 − x12x22) More succinctly Z 1 µ(A) := 2 dx A (det(x)) Haar measure on GL(n, R)

From properties of the integral, µ is countably additive, regular and non-zero. To see that µ is left-invariant, let   g11 g12 + g = ∈ GL(2, R) g21 g22

4 4 and define T : R → R by T (u) = gu (matrix multiplication). Then by definition of µ Z 1 µ(gA) = 2 dx11dx12dx21dx22 T (A) (det(x))

so by change of variables with x = T (u) = gu

Z 1 µ(gA) = 2 Jac(T ) du11du12du21du22 A (det(gu)) Haar measure on GL(n, R) A computation shows that Jac(T ) = (det(g))2, hence Z 1 2 µ(gA) = 2 2 (det(g)) du11du12du21du22 A (det(g)) (det(u)) Z 1 = 2 du A (det(u)) = µ(A)

So µ is left-invariant. A similar computation shows that µ is also right-invariant.

Therefore a (left- and right-invariant) Haar measure on + G = GL(2, R) is given by Z 1 µ(A) := 2 dx A (det(x)) A more sophisticated approach

When a locally compact group G is a smooth manifold, we can integrate over G and express the change-of-variables rule using tensors, pullbacks, differential forms, etc. At this level, the formal symbols of integration (dx, dy etc) take on real meanings! A Haar measure on G can then be defined using an integral with an integrand that ensures left-invariance. Some good references for integration on manifolds are:

I Guillemin and Pollack, Differential Topology

I Milnor, Topology from the Differential Viewpoint

I Spivak, Calculus on Manifolds