The Banach{Tarski Paradox and Amenability Lecture 9: Haar Measure 25 August 2011 Topological groups Definition A topological group 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 2 G, are continuous functions. Examples × 1. The multiplicative group R of non-zero real numbers is a topological group, 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) = fg 2 Mn;n(R) : det(g) 6= 0g 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 2 X , there is a compact set K and an open set U such that x 2 U ⊆ K. Definition A locally compact topological group or locally compact group is a topological group G which is locally compact as a topological space. Examples × 1. R is a locally compact group. 2. GL(n; R) is a locally compact group, since every open subset n2 of R is locally compact in the inherited topology. Haar measure Definition Let G be a locally compact group. A (left-invariant) Haar measure on G is a regular Borel measure such that for all g 2 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 2 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 2 X have open neighbourhoods U; U0 such that there are homeomorphisms ' : U ! V and '0 : U0 ! V 0, respectively, where V and V 0 are n open subsets of R . Then the function 0 −1 0 0 0 ' ◦ ' j'(U\U0) : '(U \ U ) ! ' (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 determinant 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 integral Z 1 µ(A) := dx A jxj 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 jxj a jλuj a juj 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 calculus 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 subgroup + GL(2; R) = fg 2 GL(n; R) : det(g) > 0g + 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 = 2 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.