Haar measure on is defined up to a constant factor by requirement of being left and right invariant
(1)
where is an arbitrary unitary matrix [GW80; SS97; Ton18]. By default I choose constant factor such that
(2)
For examples consult [Wil77].
For any other compact group Haar measure is defined in completely analagous way.
References
[SS97]
G. W. Semenoff and R. J. Szabo, “Fermionic matrix models,” Int. J. Modern Phys. A, vol. 12, no. 12, pp. 2135–2291, 1997, doi: 10.1142/s0217751x97001328. arXiv: hep-th/9605140.
[GW80]
D. J. Gross and E. Witten, “Possible third-order phase transition in the large-\(N\) lattice gauge theory,” Phys. Rev. D, vol. 21, no. 2, pp. 446–453, 1980, doi: 10.1103/physrevd.21.446.
[Ton18]
D. Tong, “Gauge theory,” 2018. Available: https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf