Throughout this paper G
denotes a locally compact group and {H_{n}} denotes an increasing sequence of closed
subgroups of G whose union H is dense in G. For each n,Δ_{n} denotes the
modular function on H_{n} and Δ denotes the modular function on G. Then
lim_{n}Δ_{n}(x) = Δ(x) for each x ∈ H. For each n,λ_{n} denotes a left Haar measure on
H_{n} 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
Δ_{n} is the restriction of Δ to H_{n} for all sufficiently large n, then there is a
“normalizing” sequence {α_{n}} of positive numbers such that for every f in
L_{1}(G,λ)
 (1) 
for λlocally almost all x in G. The hypotheses regarding the Δ_{n}’s and Δ hold in all
cases known to the authors. In particular, they hold if the H_{n}’s are unimodular
(hence if they are Abelian, compact, or discrete) or if the H_{n}’s are open subgroups or
normal subgroups. If G is the compact group 0,1 with addition modulo 1, if the
H_{n}’s are the finite groups {k2^{−n} : 0 ≦ k ≦ 2^{n} − 1} with counting measure λ_{n}, and if
α_{n} = 2^{−n}, then the left side of (1) is a Riemann sum and (1) becomes Jessen’s
theorem.
