Haar measure

Let ${\displaystyle (G,\cdot )}$ be a locally compact Hausdorff topological group. The ${\displaystyle \sigma }$-algebra generated by all open subsets of ${\displaystyle G}$ is called the Borel algebra. An element of the Borel algebra is called a Borel set. If ${\displaystyle g}$ is an element of ${\displaystyle G}$ and ${\displaystyle S}$ is a subset of ${\displaystyle G}$, then we define the left and right translates of ${\displaystyle S}$ by g as follows:

• Left translate:
  $${\displaystyle gS=\{g\cdot s\,:\,s\in S\}.}$$

  
• Right translate:
  $${\displaystyle Sg=\{s\cdot g\,:\,s\in S\}.}$$

  

Left and right translates map Borel sets onto Borel sets.

A measure ${\displaystyle \mu }$ on the Borel subsets of ${\displaystyle G}$ is called left-translation-invariant if for all Borel subsets ${\displaystyle S\subseteq G}$ and all ${\displaystyle g\in G}$ one has

${\displaystyle \mu (gS)=\mu (S).}$

A measure ${\displaystyle \mu }$ on the Borel subsets of ${\displaystyle G}$ is called right-translation-invariant if for all Borel subsets ${\displaystyle S\subseteq G}$ and all ${\displaystyle g\in G}$ one has

${\displaystyle \mu (Sg)=\mu (S).}$

There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure ${\displaystyle \mu }$ on the Borel subsets of ${\displaystyle G}$ satisfying the following properties:

• The measure ${\displaystyle \mu }$ is left-translation-invariant: ${\displaystyle \mu (gS)=\mu (S)}$ for every ${\displaystyle g\in G}$ and all Borel sets ${\displaystyle S\subseteq G}$.
• The measure ${\displaystyle \mu }$ is finite on every compact set: ${\displaystyle \mu (K)<\infty }$ for all compact ${\displaystyle K\subseteq G}$.
• The measure ${\displaystyle \mu }$ is outer regular on Borel sets ${\displaystyle S\subseteq G}$:
  $${\displaystyle \mu (S)=\inf\{\mu (U):S\subseteq U,U{\text{ open}}\}.}$$

  
• The measure ${\displaystyle \mu }$ is inner regular on open sets ${\displaystyle U\subseteq G}$:
  $${\displaystyle \mu (U)=\sup\{\mu (K):K\subseteq U,K{\text{ compact}}\}.}$$

  

Such a measure on ${\displaystyle G}$ is called a left Haar measure. It can be shown as a consequence of the above properties that ${\displaystyle \mu (U)>0}$ for every non-empty open subset ${\displaystyle U\subseteq G}$. In particular, if ${\displaystyle G}$ is compact then ${\displaystyle \mu (G)}$ is finite and positive, so we can uniquely specify a left Haar measure on ${\displaystyle G}$ by adding the normalization condition ${\displaystyle \mu (G)=1}$.

In complete analogy, one can also prove the existence and uniqueness of a right Haar measure on ${\displaystyle G}$. The two measures need not coincide.

Some authors define a Haar measure on Baire sets rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular. Halmos[3] rather confusingly uses the term "Borel set" for elements of the ${\displaystyle \sigma }$-ring generated by compact sets, and defines Haar measures on these sets.

The left Haar measure satisfies the inner regularity condition for all ${\displaystyle \sigma }$-finite Borel sets, but may not be inner regular for all Borel sets. For example, the product of the unit circle (with its usual topology) and the real line with the discrete topology is a locally compact group with the product topology and a Haar measure on this group is not inner regular for the closed subset ${\displaystyle \{1\}\times [0,1]}$. (Compact subsets of this vertical segment are finite sets and points have measure ${\displaystyle 0}$, so the measure of any compact subset of this vertical segment is ${\displaystyle 0}$. But, using outer regularity, one can show the segment has infinite measure.)

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil.[4] Weil's proof used the axiom of choice and Henri Cartan furnished a proof that avoided its use.[5] Cartan's proof also establishes the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by Alfsen in 1963.[6] The special case of invariant measure for second-countable locally compact groups had been shown by Haar in 1933.[1]

• If ${\displaystyle G}$ is a discrete group, then the compact subsets coincide with the finite subsets, and a (left and right invariant) Haar measure on ${\displaystyle G}$ is the counting measure.
• The Haar measure on the topological group ${\displaystyle (\mathbb {R} ,+)}$ that takes the value ${\displaystyle 1}$ on the interval ${\displaystyle [0,1]}$ is equal to the restriction of Lebesgue measure to the Borel subsets of ${\displaystyle \mathbb {R} }$. This can be generalized to ${\displaystyle (\mathbb {R} ^{n},+).}$
• In order to define a Haar measure ${\displaystyle \mu }$ on the circle group ${\displaystyle \mathbb {T} }$, consider the function ${\displaystyle f}$ from ${\displaystyle [0,2\pi ]}$ onto ${\displaystyle \mathbb {T} }$ defined by ${\displaystyle f(t)=(\cos(t),\sin(t))}$. Then ${\displaystyle \mu }$ can be defined by
  $${\displaystyle \mu (S)={\frac {1}{2\pi }}m(f^{-1}(S)),}$$

  

  where ${\displaystyle m}$ is the Lebesgue measure on ${\displaystyle [0,2\pi ]}$. The factor ${\displaystyle (2\pi )^{-1}}$ is chosen so that ${\displaystyle \mu (\mathbb {T} )=1}$.

• If ${\displaystyle G}$ is the group of positive real numbers under multiplication then a Haar measure ${\displaystyle \mu }$ is given by
  $${\displaystyle \mu (S)=\int _{S}{\frac {1}{t}}\,dt}$$

  

  for any Borel subset ${\displaystyle S}$ of positive real numbers. For example, if ${\displaystyle S}$ is taken to be an interval ${\displaystyle [a,b]}$, then we find ${\displaystyle \mu (S)=\log(b/a)}$. Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number ${\displaystyle g}$, resulting in ${\displaystyle gS}$ being the interval ${\displaystyle [g\cdot a,g\cdot b].}$ Measuring this new interval, we find ${\displaystyle \mu (gS)=\log((g\cdot b)/(g\cdot a))=\log(b/a)=\mu (S).}$

• If ${\displaystyle G}$ is the group of nonzero real numbers with multiplication as operation, then a Haar measure ${\displaystyle \mu }$ is given by
  $${\displaystyle \mu (S)=\int _{S}{\frac {1}{|t|}}\,dt}$$

  

  for any Borel subset ${\displaystyle S}$ of the nonzero reals.

• For the general linear group ${\displaystyle G=GL(n,\mathbb {R} )}$, any left Haar measure is a right Haar measure and one such measure ${\displaystyle \mu }$ is given by
  $${\displaystyle \mu (S)=\int _{S}{1 \over |\det(X)|^{n}}\,dX}$$

  

  where ${\displaystyle dX}$ denotes the Lebesgue measure on ${\displaystyle \mathbb {R} ^{n^{2}}}$ identified with the set of all ${\displaystyle n\times n}$-matrices. This follows from the change of variables formula.

• Generalizing the previous three examples, if the group ${\displaystyle G}$ is represented as an open submanifold of ${\displaystyle \mathbb {R} ^{n}}$ with smooth group operations, then a left Haar measure on ${\displaystyle G}$ is given by ${\displaystyle {\frac {1}{|J_{(x\cdot )}(e_{1})|}}d^{n}x}$, where ${\displaystyle e_{1}}$ is the group identity element of ${\displaystyle G}$, ${\displaystyle J_{(x\cdot )}(e_{1})}$ is the Jacobian determinant of left multiplication by ${\displaystyle x}$ at ${\displaystyle e_{1}}$, and ${\displaystyle d^{n}x}$ is the Lebesgue measure on ${\displaystyle \mathbb {R} ^{n}}$. This follows from the change of variables formula. A right Haar measure is given in the same way, except with ${\displaystyle J_{(\cdot x)}(e_{1})}$ being the Jacobian of right multiplication by ${\displaystyle x}$.
• For the orthogonal group ${\displaystyle G=O(n)}$, its Haar measure can be constructed as follows (as the distribution of a random variable). First sample ${\displaystyle A\sim N(0,1)^{n\times n}}$, that is, a matrix with all entries being IID samples of the normal distribution with mean zero and variance one. Next use Gram–Schmidt process on the matrix; the resulting random variable takes values in ${\displaystyle O(n)}$ and it is distributed according to the probability Haar measure on that group.[7] Since the special orthogonal group ${\displaystyle SO(n)}$ is an open subgroup of ${\displaystyle O(n)}$ the restriction of Haar measure of ${\displaystyle O(n)}$ to ${\displaystyle SO(n)}$ gives a Haar measure on ${\displaystyle SO(n)}$ (in random variable terms this means conditioning the determinant to be 1, an event of probability 1/2).
• The same method as for ${\displaystyle O(n)}$ can be used to construct the Haar measure on the unitary group ${\displaystyle U(n)}$. For the special unitary group ${\displaystyle G=SU(n)}$ (which has measure 0 in ${\displaystyle U(n)}$), its Haar measure can be constructed as follows. First sample ${\displaystyle A}$ from the Haar measure (normalized to one, so that it's a probability distribution) on ${\displaystyle U(n)}$, and let ${\displaystyle e^{i\theta }=\det A}$, where ${\displaystyle \theta }$ may be any one of the angles, then independently sample ${\displaystyle k}$ from the uniform distribution on ${\displaystyle \{1,...,n\}}$. Then ${\displaystyle e^{-i{\frac {\theta +2\pi k}{n}}}A}$ is distributed as the Haar measure on ${\displaystyle SU(n)}$.
• Let ${\displaystyle G}$ be the set of all affine linear transformations ${\displaystyle A:\mathbb {R} \to \mathbb {R} }$ of the form ${\displaystyle r\mapsto xr+y}$ for some fixed ${\displaystyle x,y\in \mathbb {R} }$ with ${\displaystyle x>0.}$ Associate with ${\displaystyle G}$ the operation of function composition ${\displaystyle \circ }$, which turns ${\displaystyle G}$ into a non-abelian group. ${\displaystyle G}$ can be identified with the right half plane ${\displaystyle (0,\infty )\times \mathbb {R} =\left\{(x,y)~:~x,y\in \mathbb {R} ,x>0\right\}}$ under which the group operation becomes ${\displaystyle (s,t)\circ (u,v)=(su,sv+t).}$ A left-invariant Haar measure ${\displaystyle \mu _{L}}$ (respectively, a right-invariant Haar measure ${\displaystyle \mu _{R}}$) on ${\displaystyle G=(0,\infty )\times \mathbb {R} }$ is given by
  $${\displaystyle \mu _{L}(S)=\int _{S}{\frac {1}{x^{2}}}\,dx\,dy}$$

  
      and    
  $${\displaystyle \mu _{R}(S)=\int _{S}{\frac {1}{x}}\,dx\,dy}$$

  

  for any Borel subset ${\displaystyle S}$ of ${\displaystyle G=(0,\infty )\times \mathbb {R} .}$ This is because if ${\displaystyle S\subseteq (0,\infty )\times \mathbb {R} }$ is an open subset then for ${\displaystyle (s,t)\in G}$ fixed, integration by substitution gives

  $${\displaystyle \mu _{L}((s,t)\circ S)=\int _{(s,t)\circ S}{\frac {1}{x^{2}}}\,dx\,dy=\int _{S}{\frac {1}{(su)^{2}}}|(s)(s)-(0)(0)|\,du\,dv=\mu _{L}(S)}$$

  

  while for ${\displaystyle (u,v)\in G}$ fixed,

  $${\displaystyle \mu _{R}(S\circ (u,v))=\int _{S\circ (u,v)}{\frac {1}{x}}\,dx\,dy=\int _{S}{\frac {1}{su}}|(u)(1)-(v)(0)|\,ds\,dt=\mu _{R}(S).}$$

  
• On any Lie group of dimension ${\displaystyle d}$ a left Haar measure can be associated with any non-zero left-invariant ${\displaystyle d}$-form ${\displaystyle \omega }$, as the Lebesgue measure ${\displaystyle |\omega |}$; and similarly for right Haar measures. This means also that the modular function can be computed, as the absolute value of the determinant of the adjoint representation.
• The unit hyperbola ${\displaystyle \{(x,y):x^{2}-y^{2}=1,x>0\}}$ can be taken as a group under multiplication defined as with split-complex numbers ${\displaystyle z=x+yj,\ \ j^{2}=+1.}$ The usual area measure in the crescent ${\displaystyle C=\{(x,y):|y|<x,\ \ x^{2}-y^{2}<1\}}$ serves to define hyperbolic angle as the area of its hyperbolic sector. The Haar measure of the unit hyperbola is generated by the hyperbolic angle of segments on the hyperbola. For instance, a measure of one unit is given by the segment running from (1,1) to (e,1/e), where e is Euler's number. Hyperbolic angle has been exploited in mathematical physics with rapidity standing in for classical velocity.
• If ${\displaystyle G}$ is the group of non-zero quaternions, then ${\displaystyle G}$ can be seen as an open subset of ${\displaystyle \mathbb {R} ^{4}}$. A Haar measure ${\displaystyle \mu }$ is given by
  $${\displaystyle \mu (S)=\int _{S}{\frac {1}{(x^{2}+y^{2}+z^{2}+w^{2})^{2}}}\,dx\,dy\,dz\,dw}$$

  

  where ${\displaystyle dx\wedge dy\wedge dz\wedge dw}$ denotes the Lebesgue measure in ${\displaystyle \mathbb {R} ^{4}}$ and ${\displaystyle S}$ is a Borel subset of ${\displaystyle G}$.

• If ${\displaystyle G}$ is the additive group of ${\displaystyle p}$-adic numbers for a prime ${\displaystyle p}$, then a Haar measure is given by letting ${\displaystyle a+p^{n}O}$ have measure ${\displaystyle p^{-n}}$, where ${\displaystyle O}$ is the ring of ${\displaystyle p}$-adic integers.

It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ${\displaystyle \nu }$ satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure ${\displaystyle \mu }$. The left and right Haar measures are the same only for so-called unimodular groups (see below). It is quite simple, though, to find a relationship between ${\displaystyle \mu }$ and ${\displaystyle \nu }$.

Indeed, for a Borel set ${\displaystyle S}$, let us denote by ${\displaystyle S^{-1}}$ the set of inverses of elements of ${\displaystyle S}$. If we define

${\displaystyle \mu _{-1}(S)=\mu (S^{-1})\quad }$

then this is a right Haar measure. To show right invariance, apply the definition:

${\displaystyle \mu _{-1}(Sg)=\mu ((Sg)^{-1})=\mu (g^{-1}S^{-1})=\mu (S^{-1})=\mu _{-1}(S).\quad }$

Because the right measure is unique, it follows that ${\displaystyle \mu _{-1}}$ is a multiple of ${\displaystyle \nu }$ and so

${\displaystyle \mu (S^{-1})=k\nu (S)\,}$

for all Borel sets ${\displaystyle S}$, where ${\displaystyle k}$ is some positive constant.

If the locally compact group ${\displaystyle G}$ acts transitively on a homogeneous space ${\displaystyle G/H}$, one can ask if this space has an invariant measure, or more generally a semi-invariant measure with the property that ${\displaystyle \mu (gS)=\chi (g)\mu (S)}$ for some character ${\displaystyle \chi }$ of ${\displaystyle G}$. A necessary and sufficient condition for the existence of such a measure is that the restriction ${\displaystyle \chi |_{H}}$ is equal to ${\displaystyle \Delta |_{H}/\delta }$, where ${\displaystyle \Delta }$ and ${\displaystyle \delta }$ are the modular functions of ${\displaystyle G}$ and ${\displaystyle H}$ respectively.[8] In particular an invariant measure on ${\displaystyle G/H}$ exists if and only if the modular function ${\displaystyle \Delta }$ of ${\displaystyle G}$ restricted to ${\displaystyle H}$ is the modular function ${\displaystyle \delta }$ of ${\displaystyle H}$.

Using the general theory of Lebesgue integration, one can then define an integral for all Borel measurable functions ${\displaystyle f}$ on ${\displaystyle G}$. This integral is called the Haar integral and is denoted as:

${\displaystyle \int f(x)\,d\mu (x)}$

where ${\displaystyle \mu }$ is the Haar measure.

One property of a left Haar measure ${\displaystyle \mu }$ is that, letting ${\displaystyle s}$ be an element of ${\displaystyle G}$, the following is valid:

${\displaystyle \int _{G}f(sx)\ d\mu (x)=\int _{G}f(x)\ d\mu (x)}$

for any Haar integrable function ${\displaystyle f}$ on ${\displaystyle G}$. This is immediate for indicator functions:

${\displaystyle \int {\mathit {1}}_{A}(tg)\,d\mu =\int {\mathit {1}}_{t^{-1}A}(g)\,d\mu =\mu (t^{-1}A)=\mu (A)=\int {\mathit {1}}_{A}(g)\,d\mu ,}$

which is essentially the definition of left invariance.

In the same issue of Annals of Mathematics and immediately after Haar's paper, the Haar theorem was used to solve Hilbert's fifth problem restricted to compact groups by John von Neumann.[9]

Unless ${\displaystyle G}$ is a discrete group, it is impossible to define a countably additive left-invariant regular measure on all subsets of ${\displaystyle G}$, assuming the axiom of choice, according to the theory of non-measurable sets.

In 1936, André Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain separating property,[3] then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as the Haar measure on this completion.

• Invariant measure
• Pontryagin duality
• Riesz–Markov–Kakutani representation theorem

• Diestel, Joe; Spalsbury, Angela (2014), The joys of Haar measure, Graduate Studies in Mathematics, vol. 150, Providence, RI: American Mathematical Society, ISBN 978-1-4704-0935-7, MR 3186070
• Loomis, Lynn (1953), An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., hdl:2027/uc1.b4250788.
• Hewitt, Edwin; Ross, Kenneth A. (1963), Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations., Die Grundlehren der mathematischen Wissenschaften, vol. 115, Berlin-Göttingen-Heidelberg: Springer-Verlag, MR 0156915
• Nachbin, Leopoldo (1965), The Haar Integral, Princeton, NJ: D. Van Nostrand
• André Weil, Basic Number Theory, Academic Press, 1971.

• The existence and uniqueness of the Haar integral on a locally compact topological group - by Gert K. Pedersen
• On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups - by Simon Rubinstein-Salzedo