Vol. 20, No. 1, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Jessen’s theorem on Riemann sums for locally compact groups

Kenneth Allen Ross and Karl Robert Stromberg

Vol. 20 (1967), No. 1, 135–147
Abstract

Throughout this paper G denotes a locally compact group and {Hn} denotes an increasing sequence of closed subgroups of G whose union H is dense in G. For each n,Δn denotes the modular function on Hn and Δ denotes the modular function on G. Then limnΔn(x) = Δ(x) for each x H. For each n,λn denotes a left Haar measure on Hn and λ denotes a left Haar measure on G. For a function f on G and an x in G,xf denotes the function xf(y) = f(xy). The main theorem states that if Δn is the restriction of Δ to Hn for all sufficiently large n, then there is a “normalizing” sequence {αn} of positive numbers such that for every f in L1(G,λ)

     ∫          ∫
m α      f dλ =   f dλ
n n  Hnx    n   e
(1)

for λ-locally almost all x in G. The hypotheses regarding the Δn’s and Δ hold in all cases known to the authors. In particular, they hold if the Hn’s are unimodular (hence if they are Abelian, compact, or discrete) or if the Hn’s are open subgroups or normal subgroups. If G is the compact group [0,1[ with addition modulo 1, if the Hn’s are the finite groups {k2n : 0 k 2n 1} with counting measure λn, and if αn = 2n, then the left side of (1) is a Riemann sum and (1) becomes Jessen’s theorem.

Mathematical Subject Classification
Primary: 22.20
Milestones
Received: 18 October 1965
Published: 1 January 1967
Authors
Kenneth Allen Ross
Karl Robert Stromberg