On the history of Haar measure

Question

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.

Answer 1

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."

Answer 2

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