# On the history of Haar measure

Haar measure is a well-known concept in measure theory.

Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.

I am looking for a good reference on the history of Haar measure.

Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt:

"Invariant integration on one or another special class of groups has long been known and used. A detailed computation of the invariant integral on $$\mathfrak{SD}(n)$$ was given in 1897 by HURWITZ [1]. SCHUR and FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1] computed and applied intensively the invariant integrals for $$\mathfrak{SD}(n)$$ and $$\mathfrak{D}(n)$$. WEYL in [1] computed the invariant integrals for $$\mathfrak{U}(n)$$, $$\mathfrak{SD}(n)$$, the unitary subgroup of the symplectic group, and [more or less explicitly] for certain other compact Lie groups. WEYL and PETER in [1] showed the existence of an invariant integral for any compact Lie group.

The decisive step in founding modern harmonic analysis was taken by A. HAAR [3] in 1933. He proved directly the existence [but not the uniqueness] of left Haar measure on a locally compact group with a countable open basis. His construction was reformulated in t erms of linear functionals and extended to arbitrary locally compact groups by A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also that HAAR's construction can be extended to all locally compact groups. Theorem ( 15. S) as stated is thus due to WEIL. The proof we present is due to H. CARTAN [1].

For an arbitrary compact group G, VON NEUMANN [5] proved the existence and uniqueness of the Haar integral, as well as its two-sided and inversion invariance. In [6], VON NEUMANN proved the uniqueness of left Haar measure for locally compact G with a countable open basis; a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38, proved the uniqueness of the left Haar integral for all locally compact groups."

Try these references:

• Section 7.5 of History of Topology, edited by I. M. James.

• Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen